#696685

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, 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,c^2=f,d^2=fg,e^2=g,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, 603881);