#8105457

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5013 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)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,27)(20,24)(21,26)(22,25)(23,29)(28,30)(31,32);(2,5)(3,7)(9,11)(10,12)(13,14)(15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,15)(13,16)(17,20,22,26)(18,24,25,21)(19,28,29,32)(23,31,27,30);(2,5)(6,8)(11,14)(12,16)(18,25)(20,26)(23,27)(28,32);(10,15)(11,14)(19,29)(23,27)(28,32)(30,31);(17,21,22,24)(18,26,25,20)(19,30,29,31)(23,32,27,28)

Polycyclic presentation:

2^9|c^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8012653);