#7634213

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

4899 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)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,29,23)(20,25,21,24)(28,31,30,32);(2,5)(6,8)(11,15)(13,16);(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,31)(26,27);(11,15)(13,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,27,32);(10,14)(13,16)(17,21,22,20)(18,25,26,24)(19,28,29,30)(23,31,27,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,d^2=h,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, 7541409);