#8341819

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

4680 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gh,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8249015);