#8699317

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

4771 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8606513);