#472326

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

922 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)(11,20)(12,26)(13,30)(14,18)(15,19)(16,23)(17,27)(21,24)(22,36)(25,32)(28,33)(29,40)(31,38)(34,37)(35,39);(2,5)(6,8)(9,11)(10,16,19,32)(12,22)(13,23)(14,25)(15,24)(17,31,33,36)(18,20,30,21)(26,34,40,35)(27,37)(28,39)(29,38);(3,7)(6,8)(9,10)(11,21,24,20)(12,26)(13,30)(14,18)(15,19)(16,23,32,25)(17,27)(22,31,38,36)(28,33)(29,40)(34,37,35,39);(1,3,4,7)(2,6,5,8)(9,12,13,27,15,29,14,28)(10,17,18,26,19,33,30,40)(11,22,23,37,24,38,25,39)(16,31,20,34,32,36,21,35)

Polycyclic presentation:

2^9|b^2=gh,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379522);