#8105580

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

5757 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)(3,4,8,6)(9,10)(11,14)(12,16)(13,15)(17,18,23,20)(19,26,28,24)(21,30,22,25)(27,31,29,32);(1,3,5,8)(2,6,7,4)(9,11,13,16)(10,14,15,12)(17,19)(18,24)(20,26)(21,27)(22,29)(23,28)(25,31)(30,32);(2,7)(4,6)(9,12,13,14)(10,11,15,16)(17,20,23,18)(19,26,28,24)(21,25,22,30)(27,31,29,32);(1,4,5,6)(2,3,7,8)(11,16)(12,14)(17,21,23,22)(18,25,20,30)(19,27,28,29)(24,31,26,32);(2,7)(3,8)(10,15)(11,16)(19,28)(24,26)(27,29)(31,32);(17,22,23,21)(18,25,20,30)(19,29,28,27)(24,31,26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^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, 8012776);