#388810

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

987 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)(3,5)(4,6)(7,8)(10,15,18,28)(11,19)(12,24,26,27)(16,20,33,30)(21,37,35,39)(22,23)(32,38)(36,40);(1,3,4,7)(2,5,6,8)(9,10,14,18)(11,20,19,30)(12,25,26,31)(13,28,29,15)(16,22,33,23)(17,27,34,24)(21,38,35,32)(36,37,40,39);(2,6)(5,8)(9,11,13,22,14,19,29,23)(10,16,28,20,18,33,15,30)(12,21,27,37,26,35,24,39)(17,32,25,36,34,38,31,40);(2,6)(5,8)(9,12)(10,17)(11,21)(13,27)(14,26)(15,31)(16,32)(18,34)(19,35)(20,36)(22,37)(23,39)(24,29)(25,28)(30,40)(33,38)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=eh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296006);