#5966800

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=g,d^2=gh,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, 5873996);