#647371

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

1582 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=f,d^2=g,e^2=hi,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, 554567);