#735452

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1566 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)(3,8,4,7)(10,13)(14,16)(17,18,21,20)(19,23,24,22)(25,26,30,29)(27,33,35,31)(28,36,38,32)(34,39,37,40);(17,18,21,20)(19,23,24,22)(26,29)(31,33)(32,36)(39,40);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,20,23)(25,27)(26,31)(28,37)(29,33)(30,35)(32,40)(34,38)(36,39);(1,4,6,3)(2,8,5,7)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,40)(35,37);(1,5,6,2)(3,8,4,7)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)(25,29,30,26)(27,33,35,31)(28,36,38,32)(34,40,37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=fhi,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, 642648);