#390796

order	512
nilpotency class	3
isoclinism	isoclinic class of #219706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

897 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)(4,7)(5,8)(9,10)(11,19,22,33)(12,21,24,29)(13,25)(14,17)(15,18)(16,28)(20,35,37,31)(23,38,40,26)(27,39)(30,34)(32,36);(1,3,6,2)(4,8,5,7)(9,11,15,22)(10,12,18,24)(13,26,28,38)(14,29,30,21)(16,31,25,35)(17,33,34,19)(20,36,37,27)(23,39,40,32);(1,4,6,5)(2,7,3,8)(9,12)(10,11,18,22)(13,23)(14,19)(15,24)(16,20,25,37)(17,21,34,29)(26,39,38,32)(27,31)(28,40)(30,33)(35,36);(1,5,6,4)(2,7,3,8)(9,13,14,27,15,28,30,36)(10,16,17,32,18,25,34,39)(11,20,21,26,22,37,29,38)(12,23,19,31,24,40,33,35)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=efgh,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,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, 297992);