#380899

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,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, 288095);