#2568629

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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

1436 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=fi,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2475825);