#363015

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

680 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)(12,24,26,16)(13,17)(14,19)(15,23)(18,27)(21,34,35,33)(22,32)(25,31,39,30)(28,36,38,40)(29,37);(2,5)(3,7)(9,11,13,23,14,22,27,37)(10,15,18,20,19,29,17,32)(12,21,25,36,26,35,39,40)(16,28,31,33,24,38,30,34);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,21)(13,25)(14,26)(15,28)(17,30)(18,31)(19,24)(20,33)(22,35)(23,36)(27,39)(29,38)(32,34)(37,40);(2,5)(6,8)(10,17,19,18)(11,22)(15,20,29,32)(16,30,24,31)(21,35)(23,37)(28,33,38,34)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=g,d^2=f,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, 270211);