#390638

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

759 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=gh,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, 297834);