#8722984

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

4770 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8630180);