#3587127

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1696 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)(10,14)(12,16)(17,18,21,19)(20,23,24,22)(25,26,29,30)(27,31,34,38)(28,32,37,36)(33,39,35,40);(3,8)(5,7)(10,14)(11,15)(17,18,21,19)(20,22,24,23)(26,30)(31,38)(32,36)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19,21,18)(20,23,24,22)(25,27,29,34)(26,31,30,38)(28,35,37,33)(32,40,36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,20,21,24)(18,22,19,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,40)(34,38)(35,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(25,28)(26,32)(27,33)(29,37)(30,36)(31,39)(34,35)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=ghi,d^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3494323);