#936765

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

1586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(3,7)(6,8)(9,10,13,11)(12,15,16,14)(17,18,20,22)(19,21,24,23)(25,26,30,31)(27,28,32,36)(29,37,38,33)(34,40,39,35);(2,5)(6,8)(17,18,20,22)(19,23,24,21)(26,31)(27,32)(33,37)(34,39);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,20,24)(18,21,22,23)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,37)(38,40);(25,29,30,38)(26,33,31,37)(27,34,32,39)(28,35,36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=i,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 843961);