#8689222

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

4912 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,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, 8596418);