#8613273

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

4998 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8520469);