#390639

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

807 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ghi,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, 297835);