#5810400

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

1510 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)(3,8)(4,7)(5,6)(9,10,13,16)(11,12,15,14)(19,20)(22,23)(26,30)(28,36)(31,33)(34,40);(2,5)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18,21,24)(19,22,20,23)(26,30)(28,36)(31,33)(34,40);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,4,6,3)(2,7,5,8)(9,11)(10,12)(13,15)(14,16)(17,19,21,20)(18,22,24,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,5,6,2)(3,8,4,7)(9,12)(10,15)(11,16)(13,14)(17,20,21,19)(18,23,24,22)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=ghi,d^2=fh,e^2=fh,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, 5717596);