#8424957

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

4823 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332153);