#8398675

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

4845 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=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, 8305871);