#7634565

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

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,e^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, 7541761);