#453915

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

646 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)(9,10,15,19,16,20,14,13)(11,21,30,17,25,29,24,23)(12,26,27,33,28,34,31,18)(22,32,37,35,39,36,38,40);(1,3,4,7)(2,5,6,8)(9,11,14,30,16,25,15,24)(10,17,13,29,20,23,19,21)(12,22,27,38,28,39,31,37)(18,32,26,40,33,36,34,35);(9,12,16,28)(10,18,20,33)(11,22)(13,26,19,34)(14,27,15,31)(17,32)(21,35)(23,36)(24,37)(25,39)(29,40)(30,38);(2,6)(5,8)(9,13,14,20,16,19,15,10)(11,23,24,29,25,17,30,21)(12,26,27,33,28,34,31,18)(22,36,37,40,39,32,38,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361111);