#8697515

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

5053 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=ghi,f^2=g,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, 8604711);