#8758030

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

5399 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8665226);