#8108788

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

5350 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=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, 8015984);