#363078

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

685 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)(4,5)(7,8)(9,10)(11,20,23,34)(12,24,26,16)(13,18)(14,19)(15,22,31,38)(17,25,32,39)(21,35)(27,28)(29,36)(30,37)(33,40);(2,5)(3,7)(9,11,13,22,14,23,27,38)(10,15,28,34,19,31,18,20)(12,21,25,36,26,37,39,40)(16,29,32,35,24,33,17,30);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,21)(13,25)(14,26)(15,29)(17,18)(19,24)(20,30)(22,36)(23,37)(27,39)(28,32)(31,33)(34,35)(38,40);(2,5)(6,8)(9,12)(10,17,19,32)(11,21)(13,25)(14,26)(15,30,31,35)(16,18,24,28)(20,33,34,29)(22,36)(23,37)(27,39)(38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=g,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,h^c=hi,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, 270274);