#380898

order	512
nilpotency class	3
isoclinism	isoclinic class of #219518
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
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

954 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)(11,22,26,32)(12,18)(13,19)(14,20)(15,21)(16,24,17,25)(23,38,40,34)(27,33,39,31)(28,35)(29,36)(30,37);(2,5)(6,8)(9,11)(10,16,21,17)(12,23)(13,24)(14,25)(15,26)(18,31,37,33)(19,22,20,32)(27,28)(29,39)(30,40)(34,35,38,36);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,27)(13,25)(14,24)(15,26)(16,21)(18,34)(19,22)(20,32)(23,28)(29,40)(30,39)(31,36)(33,35)(37,38);(3,7)(6,8)(9,12,14,29,15,30,13,28)(10,18,20,36,21,37,19,35)(11,23,25,39,26,40,24,27)(16,31,32,34,17,33,22,38)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=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, 288094);