#397938

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	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1879 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^c=ei,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, 305134);