#643533

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=g,d^2=gi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 550729);