#3813430

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

1765 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ghi,d^2=fgi,e^2=hi,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, 3720626);