#5963263

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1582 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)(9,10)(11,15)(12,13)(14,16)(17,18,20,21)(19,23,24,22)(25,26)(27,34)(28,33)(29,32)(30,31)(35,40)(36,39)(37,38);(2,5)(6,8)(10,12)(11,16)(18,21)(22,23)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(11,16)(14,15)(17,19)(18,22)(20,24)(21,23)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,19)(18,22)(20,24)(21,23)(25,29,31,27)(26,32,30,34)(28,35,38,36)(33,40,37,39);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,30,31,26)(27,32,29,34)(28,37,38,33)(35,39,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=fg,e^2=fg,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, 5870459);