#6767985

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6675181);