#7000997

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	#45064	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=fi,e^2=f,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,e^c=ei,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, 6908193);