#5960494

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5867690);