#7034457

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42474	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^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=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6941653);