#395219

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 302415);