#8105566

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

5857 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18,22,25)(19,26,28,23)(20,24,30,21)(27,29,32,31);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(17,19,22,28)(18,23,25,26)(20,27,30,32)(21,29,24,31);(1,4,5,8)(2,6,3,7)(11,16)(13,15)(17,20)(18,24)(19,27)(21,25)(22,30)(23,31)(26,29)(28,32);(2,3)(4,8)(10,14)(13,15)(18,25)(20,30)(23,26)(27,32);(2,3)(6,7)(19,28)(23,26)(27,32)(29,31);(17,21,22,24)(18,20,25,30)(19,29,28,31)(23,27,26,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,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, 8012762);