#363649

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=eg,e^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, 270845);