#8395307

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

4874 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:

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

Polycyclic presentation:

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

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, 8302503);