#5894356

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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

1684 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=fghi,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5801552);