#2606854

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=g,d^2=f,e^2=fg,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, 2514050);