#391803

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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1301 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)(13,26)(15,28)(18,31)(21,27)(24,39)(29,37)(30,38)(35,40);(2,5)(3,7)(9,10)(11,19,22,32)(12,23,25,16)(13,18)(14,17)(15,21,28,27)(20,34)(24,38,39,30)(26,31)(29,35)(33,36)(37,40);(3,7)(6,8)(9,11)(10,15)(12,20)(13,27)(14,22)(16,29)(17,28)(18,32)(19,31)(21,26)(23,37)(24,40)(25,36)(30,34)(33,38)(35,39);(1,3,4,7)(2,6,5,8)(9,12,14,25)(10,16,17,23)(11,20,22,36)(13,24,26,39)(15,29,28,37)(18,30,31,38)(19,33,32,34)(21,35,27,40)

Polycyclic presentation:

2^9|b^2=h,d^2=ei,g^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, 298999);