#719094

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,c^2=fgi,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, 626290);