#644119

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

1630 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fi,d^2=hi,e^2=g,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, 551315);