#72306329

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
prime radical	#10575872	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(2,6)(3,9)(4,13)(7,14)(10,16)(11,12)(17,18)(19,28)(20,33)(21,36)(22,27)(23,25)(24,31)(26,34)(29,43)(30,44)(32,38)(35,41)(37,42)(39,46)(40,47)(45,48);(1,2)(3,10)(4,14)(5,7)(6,13)(8,12)(9,15)(11,16)(17,19,23,29)(18,24,25,38)(20,30,34,45)(21,32,22,31)(26,40,33,39)(27,43,36,28)(35,47,37,46)(41,44,42,48);(2,7)(3,11)(6,14)(9,12)(19,29)(24,38)(28,43)(30,45)(31,32)(39,40)(44,48)(46,47);(2,7)(3,11)(6,14)(9,12)(18,25)(19,29)(20,34)(27,36)(31,32)(35,37)(39,40)(44,48);(1,3,5,11)(2,8,7,15)(4,12,13,9)(6,10,14,16)(17,20)(18,26,25,33)(19,30,29,45)(21,35)(22,37)(23,34)(24,39)(27,41,36,42)(28,44)(31,46,32,47)(38,40)(43,48)

Polycyclic presentation:

2^10|b^2=j,e^2=fghij,f^2=i,g^2=j,h^2=ij,b^a=bf,c^b=ci,d^a=dj,d^b=di,e^a=eh,e^b=efgj,e^c=eij,e^d=ei,f^a=fi,f^b=fi,g^b=gj,g^e=gi,h^a=hij,h^b=hi,h^e=hi

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306341
maximal normal subgroup of	#72306341

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);