#642642

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 549838);