#7634906

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

5434 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)(10,14)(11,12)(17,18)(19,26)(20,29)(21,24)(22,25)(23,28)(27,31)(30,32);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,30)(21,27)(22,28)(24,32)(25,26)(29,31);(9,12,13,11)(10,15,14,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32);(18,25)(23,26)(24,29)(31,32);(17,21,22,20)(18,24,25,29)(19,27,28,30)(23,32,26,31)

Polycyclic presentation:

2^9|c^2=g,d^2=h,f^2=hi,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, 7542102);