#5801490

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

1665 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)(3,5)(4,6)(7,8)(9,10,13,11)(12,15,16,14)(18,20)(19,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(4,8)(6,7)(9,10,13,11)(12,15,16,14)(18,20)(22,24)(26,30)(31,33)(32,38)(39,40);(2,3)(6,7)(10,11)(14,15)(17,18,21,20)(19,24,23,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,22,20,24)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(1,4)(2,6)(3,7)(5,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,24,23,22)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgi,c^2=ghi,d^2=fg,e^2=h,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, 5708686);