#8107550

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

4863 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014746);