#9164057

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

5222 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9071253);