#370978

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1260 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278174);