#2606844

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2213 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2514040);