#8688222

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

4514 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595418);