#8700461

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

5244 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)(3,4)(9,10,14,12)(11,15,13,16)(17,18,23,24)(19,25,22,26)(20,29,30,21)(27,32,31,28);(1,2,6,5)(3,8,4,7)(9,11,14,13)(10,15,12,16)(18,24)(20,30)(25,26)(27,31);(1,3,6,4)(2,7,5,8)(9,11,14,13)(10,16,12,15)(17,19,23,22)(18,25,24,26)(20,27,30,31)(21,28,29,32);(1,4,6,3)(2,8,5,7)(9,12,14,10)(11,15,13,16)(17,20)(18,21)(19,27)(22,31)(23,30)(24,29)(25,28)(26,32);(1,5,6,2)(3,8,4,7)(9,13,14,11)(10,15,12,16)(17,21)(18,20)(19,28)(22,32)(23,29)(24,30)(25,27)(26,31);(17,22,23,19)(18,26,24,25)(20,27,30,31)(21,28,29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607657);