#391723

order	512
nilpotency class	3
isoclinism	isoclinic class of #213798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1750 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)(9,10)(11,22)(12,18)(13,19)(14,20)(15,31)(16,21)(17,25)(23,38)(24,36)(26,39)(27,34)(28,35)(29,37)(30,33)(32,40);(1,3)(2,5)(4,7)(6,8)(9,11,15,27,16,25,14,26)(10,17,20,34,21,22,31,39)(12,24,29,23,13,30,28,40)(18,33,35,38,19,36,37,32);(2,6)(5,8)(9,12,16,13)(10,18,21,19)(11,23,25,40)(14,29,15,28)(17,32,22,38)(20,37,31,35)(24,26,30,27)(33,34,36,39);(3,7)(5,8)(9,13,16,12)(10,19,21,18)(11,24,25,30)(14,28,15,29)(17,33,22,36)(20,35,31,37)(23,27,40,26)(32,39,38,34)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,g^c=gi,h^c=hi,g^d=gi,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, 298919);