#8422194

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

4786 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=h,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, 8329390);