#5805010

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1584 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fhi,c^2=ghi,d^2=fg,e^2=gh,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, 5712206);