#391807

order	512
nilpotency class	3
isoclinism	isoclinic class of #213798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1250 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)(11,16)(12,17)(13,29)(14,18)(15,19)(20,36)(21,22)(23,39)(24,31)(25,32)(26,35)(27,33)(28,34)(30,37)(38,40);(2,5)(3,7)(9,11,14,24,15,22,13,23)(10,16,18,31,19,21,29,39)(12,25,27,20,28,38,26,37)(17,32,33,36,34,40,35,30);(3,7)(6,8)(9,12,15,28)(10,17,19,34)(11,20,22,37)(13,27,14,26)(16,30,21,36)(18,35,29,33)(23,38,24,25)(31,40,39,32);(1,3,4,7)(2,6,5,8)(9,10)(11,21)(12,17)(13,18)(14,29)(15,19)(16,22)(20,36)(23,39)(24,31)(25,40)(26,33)(27,35)(28,34)(30,37)(32,38)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299003);