#8460772

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

5089 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)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(3,7)(6,8)(10,11)(14,15)(17,19,22,25)(18,23,26,21)(20,27,29,31)(24,30,32,28);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,21)(20,27)(22,25)(23,26)(24,28)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31);(1,2)(3,6)(4,5)(7,8)(17,20,22,29)(18,24,26,32)(19,27,25,31)(21,28,23,30);(17,21,22,23)(18,25,26,19)(20,28,29,30)(24,31,32,27)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8367968);