#72284552

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#233549	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=f,c^2=h,b^a=bh,c^a=cg,d^a=df,d^b=dghij,d^c=df,e^a=ei,e^b=eij,e^c=ej,f^b=fj,f^c=fi,g^a=gi,g^c=gj,g^d=gi,h^a=hj,h^d=hi

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709, #72340359, #72340377, …
maximal normal subgroup of	#72284709, #72340359, #72340377, …

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, 1, F.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);