#72257591

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×C₂²⋊#95
commutator	C₄²	C₂²×C₄²
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	C₂³×#41	D₄
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

5397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=h,e^2=ij,f^2=i,h^2=j,b^a=bf,c^b=ce,d^b=di,e^a=ei,f^a=fi,f^c=fi,g^a=gi,g^c=gij,h^b=hij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257599, #72316914, #72316970
maximal normal subgroup of	#72257599, #72316914, #72316970
maximal quotient of	#72257723

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