#8096962

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

6036 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^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, 8004158);