#986696

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

1436 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)(9,10,12,14)(11,15,16,13)(25,26)(27,33)(28,36)(29,34)(30,31)(32,37)(35,39)(38,40);(1,2,5,4)(3,6,7,8)(11,16)(13,15)(17,18,21,20)(19,23,24,22)(26,31)(27,34)(28,37)(39,40);(1,2,5,4)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,22,20,23)(25,27)(26,29)(28,38)(30,34)(31,33)(32,40)(35,37)(36,39);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,23,24,22)(25,28)(26,32)(27,35)(29,39)(30,37)(31,36)(33,40)(34,38);(1,4,5,2)(3,8,7,6)(25,29)(26,27)(28,39)(30,33)(31,34)(32,35)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=fghi,c^2=fghi,d^2=gh,e^2=gi,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, 893892);