#353539

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1786 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)(9,10,13,19,15,20,14,18)(11,21,23,16,25,31,24,32)(12,26,28,17,27,34,29,33)(22,36,38,35,37,30,39,40);(5,6,7,8)(9,11,14,24,15,25,13,23)(10,16,19,31,20,32,18,21)(12,22,29,39,27,37,28,38)(17,30,34,40,33,36,26,35);(2,4)(9,12,13,28,15,27,14,29)(10,17,18,26,20,33,19,34)(11,22,23,38,25,37,24,39)(16,30,21,35,32,36,31,40);(12,27)(17,33)(22,37)(26,34)(28,29)(30,36)(35,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260735);