#7048096

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45074	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6955292);