#7092693

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	#30733	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fgi,c^2=ghi,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6999889);