#393403

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	C₂×#388
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

916 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)(4,8)(5,7)(9,10,14,18,15,19,13,29)(11,20,24,30,21,33,23,16)(12,25,28,40,27,17,26,32)(22,36,38,31,35,34,37,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16,19,30)(12,22)(13,23)(14,24)(15,21)(17,31,25,39)(18,20,29,33)(26,37)(27,35)(28,38)(32,34,40,36);(1,4,5,3)(2,6,7,8)(11,21)(16,30)(20,33)(22,35)(23,24)(31,39)(34,36)(37,38);(9,12,13,26,15,27,14,28)(10,17,18,32,19,25,29,40)(11,22,23,37,21,35,24,38)(16,31,20,34,30,39,33,36)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,c^b=ci,g^d=gi

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, 300599);