#706300

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

1574 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)(9,10,13,16)(11,14,12,15)(19,23)(22,24)(25,26,30,31)(27,33,35,29)(28,36,39,32)(34,38,37,40);(17,18,21,20)(19,24,23,22)(26,31)(29,33)(32,36)(38,40);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,23)(18,22,20,24)(25,27)(26,29)(28,37)(30,35)(31,33)(32,40)(34,39)(36,38);(1,4,6,3)(2,8,5,7)(9,11,13,12)(10,14,16,15)(17,18,21,20)(19,22,23,24)(25,28,30,39)(26,32,31,36)(27,34,35,37)(29,38,33,40);(1,5,6,2)(3,8,4,7)(9,12,13,11)(10,15,16,14)(17,20,21,18)(19,24,23,22)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=fhi,d^2=g,e^2=fg,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, 613496);