#385476

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

750 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:

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

Polycyclic presentation:

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

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, 292672);