#496869

order	512
nilpotency class	4
isoclinism	isoclinic class of #33752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 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)(3,6)(4,5)(7,8)(9,10,15,20,13,18,30,21)(11,19,24,34,23,16,14,31)(12,25,28,36,27,40,29,33)(17,35,38,26,37,32,39,22);(2,5)(3,7)(9,11,15,24,13,23,30,14)(10,16,20,31,18,19,21,34)(12,26,28,32,27,22,29,35)(17,36,38,40,37,33,39,25);(1,3,4,7)(2,6,5,8)(9,12,13,27)(10,17,18,37)(11,22,23,26)(14,32,24,35)(15,28,30,29)(16,33,19,36)(20,38,21,39)(25,34,40,31);(2,5)(6,8)(10,18)(16,19)(17,37)(20,21)(25,40)(31,34)(33,36)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^c=fh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404065);