#1361734

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1490 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)(9,10,14,12)(11,13,16,15)(17,18,21,20)(19,22,23,24)(25,26,29,30)(27,31,34,38)(28,35,37,32)(33,40,36,39);(1,2,5,4)(3,6,7,8)(19,23)(22,24)(26,30)(31,38)(32,35)(39,40);(9,11,14,16)(10,15,12,13)(17,19)(18,22)(20,24)(21,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,13,16,15)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,4,5,2)(3,8,7,6)(9,13)(10,11)(12,16)(14,15)(17,20,21,18)(19,24,23,22)(25,28)(26,32)(27,33)(29,37)(30,35)(31,39)(34,36)(38,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=i,c^2=fg,d^2=g,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1268930);