#3517857

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1821 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)(3,5,7,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,22)(19,23,24,20)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,40)(36,37);(3,7)(5,8)(9,11,13,16)(10,14,15,12)(19,24)(20,23)(26,30)(31,38)(32,36)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,27,29,34)(26,31,30,38)(28,35,37,33)(32,40,36,39);(1,3,4,7)(2,5,6,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(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)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28,29,37)(26,32,30,36)(27,33,34,35)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=f,d^2=ghi,e^2=f,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, 3425053);