#8807303

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

4609 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=gh,f^2=gi,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, 8714499);