#8657164

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=gi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8564360);