#7634868

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

4982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,e^2=gi,f^2=gh,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, 7542064);