#391722

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×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1860 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)(11,20)(12,17)(13,18)(14,29)(15,19)(16,22)(21,36)(23,37)(24,31)(25,40)(26,33)(27,35)(28,34)(30,38)(32,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,24,15,22,13,23)(10,16,18,31,19,20,29,37)(12,25,27,21,28,39,26,38)(17,32,33,36,34,40,35,30);(2,6)(5,8)(9,12,15,28)(10,17,19,34)(11,21,22,38)(13,27,14,26)(16,30,20,36)(18,35,29,33)(23,39,24,25)(31,40,37,32);(3,7)(5,8)(11,22)(13,14)(16,20)(18,29)(25,39)(26,27)(32,40)(33,35)

Polycyclic presentation:

2^9|b^2=h,c^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, 298918);