#8898083

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

4969 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)(17,18,24,28)(19,29,32,25)(20,23,22,26)(21,31,30,27);(3,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19,24,32)(18,25,28,29)(20,30,22,21)(23,31,26,27);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,20,24,22)(18,26,28,23)(19,30,32,21)(25,27,29,31);(11,15)(13,16)(17,21)(18,27)(19,20)(22,32)(23,25)(24,30)(26,29)(28,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,22,24,20)(18,26,28,23)(19,30,32,21)(25,31,29,27);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,23,24,26)(18,20,28,22)(19,31,32,27)(21,25,30,29)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805279);