#370071

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

800 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277267);