#703984

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=fgh,e^2=fgh,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, 611180);