#363632

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
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

661 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 270828);