#7001002

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

1623 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)(17,18,22,26)(19,23,21,24)(20,25,29,32)(27,30,28,31);(1,2)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,21)(18,23,26,24)(20,27,29,28)(25,30,32,31);(9,11)(10,13)(12,16)(14,15)(18,24)(19,21)(23,26)(25,31)(27,28)(30,32);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20,22,29)(18,25,26,32)(19,27,21,28)(23,30,24,31);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,19,22,21)(18,24,26,23)(20,28,29,27)(25,30,32,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,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=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, 6908198);