#7144118

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1746 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7051314);