#726640

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

1590 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=fh,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,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, 633836);