#2723204

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1785 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=f,e^2=fghi,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, 2630400);