#2573251

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1675 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)(6,8)(9,10,12,13)(11,15,16,14)(17,18,20,22)(19,23,24,21)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(10,13)(14,15)(17,18)(19,23)(20,22)(21,24)(25,28,31,27)(26,32,29,34)(30,35,38,36)(33,40,37,39);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=fg,d^2=hi,e^2=fghi,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, 2480447);