#8422178

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

5060 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)(14,16)(17,18,24,23)(19,28,25,20)(21,30,32,26)(22,29,31,27);(1,2)(3,6)(4,7)(5,8)(11,15)(14,16)(17,19,24,25)(18,20,23,28)(21,27,32,29)(22,30,31,26);(1,3)(2,5)(4,8)(6,7)(9,10,13,12)(11,14,15,16)(17,20)(18,25)(19,23)(21,31)(22,32)(24,28)(26,27)(29,30);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,21,24,32)(18,26,23,30)(19,29,25,27)(20,31,28,22);(9,12,13,10)(11,16,15,14)(17,22,24,31)(18,27,23,29)(19,30,25,26)(20,32,28,21);(1,2)(3,5)(4,7)(6,8)(17,23,24,18)(19,28,25,20)(21,26,32,30)(22,29,31,27)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8329374);