#8912330

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

5380 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8819526);