#8424588

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

5246 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=h,e^2=h,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, 8331784);