#363840

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
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×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2285 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

750 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)(3,7)(4,6)(5,8)(9,10,15,20,16,21,14,31)(11,22,24,33,25,19,13,17)(12,26,28,18,30,35,29,36)(23,38,27,37,40,32,39,34);(1,3,4,8)(2,5,6,7)(9,11,14,13,16,25,15,24)(10,17,20,22,21,33,31,19)(12,23,28,27,30,40,29,39)(18,32,26,37,36,38,35,34);(2,6)(3,8)(9,12)(10,18,21,36)(11,23)(13,27)(14,28)(15,29)(16,30)(17,32,33,38)(19,34,22,37)(20,26,31,35)(24,39)(25,40);(2,6)(5,7)(9,13,14,25,16,24,15,11)(10,19,20,17,21,22,31,33)(12,27,28,40,30,39,29,23)(18,34,26,32,36,37,35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=eg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 271036);