#363651

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

814 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)(3,7)(4,6)(5,8)(9,10,14,19,15,20,13,16)(11,21,23,33,24,17,29,30)(12,25,28,34,26,31,27,18)(22,36,38,35,37,39,40,32);(1,3)(2,5)(4,8)(6,7)(10,16,20,19)(12,26)(17,30,21,33)(18,25,34,31)(22,37)(27,28)(32,39,35,36)(38,40);(3,8)(5,7)(9,11)(10,17)(12,22)(13,23)(14,29)(15,24)(16,30)(18,32)(19,33)(20,21)(25,39)(26,37)(27,38)(28,40)(31,36)(34,35);(2,6)(5,7)(9,12,13,27,15,26,14,28)(10,18,19,25,20,34,16,31)(11,22,23,38,24,37,29,40)(17,32,33,39,21,35,30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270847);