#7634349

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

4946 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)(11,12)(14,16)(17,18)(19,23)(20,27)(21,29)(22,25)(24,28)(26,32)(30,31);(2,5)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,27,32);(9,11,13,12)(10,14,15,16)(17,20,22,28)(18,24,25,27)(19,26,21,30)(23,31,29,32);(9,12,13,11)(10,14,15,16)(17,21,22,19)(18,23,25,29)(20,26,28,30)(24,32,27,31);(18,25)(23,29)(24,27)(31,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^c=dhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7541545);