#410209

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,g^b=gi,e^c=ei

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, 317405);