#8323166

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

4742 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,24,21)(19,25,22,31)(20,29,28,26)(23,27,30,32);(1,3)(2,6)(4,7)(5,8)(10,14)(11,16)(17,19,24,22)(18,25,21,31)(20,23,28,30)(26,32,29,27);(2,7)(4,6)(9,11)(10,12)(13,16)(14,15)(17,20)(18,26)(19,23)(21,29)(22,30)(24,28)(25,32)(27,31);(2,7)(3,8)(9,11)(10,12)(13,16)(14,15)(17,21,24,18)(19,25,22,31)(20,29,28,26)(23,32,30,27);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,22,24,19)(18,25,21,31)(20,30,28,23)(26,32,29,27);(1,4,5,6)(2,3,7,8)(17,23,24,30)(18,27,21,32)(19,28,22,20)(25,26,31,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8230362);