#8800263

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

4894 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8707459);