#8496571

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

4946 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8403767);