#8096940

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

5827 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004136);