#626941

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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

1584 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)(3,8,7,4)(10,13)(14,16)(17,18,21,20)(19,23,24,22)(26,27)(29,37)(31,33)(35,40);(19,24)(22,23)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,4,6,8)(2,7,5,3)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,5,6,2)(3,8,7,4)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=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, 534137);