#472329

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×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

891 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)(11,22)(12,25)(13,28)(14,33)(15,20)(16,21)(17,24)(18,27)(19,30)(23,37)(26,34)(29,39)(31,35)(32,40)(36,38);(2,5)(6,8)(9,11)(10,17,21,27)(12,20,22,33)(13,23)(14,24)(15,18)(16,25)(19,29,35,37)(26,28,36,40)(30,38)(31,34)(32,39);(3,7)(6,8)(9,12,14,27,16,22,15,17)(10,18,20,25,21,24,33,11)(13,29,30,26,32,37,31,36)(19,23,28,34,35,39,40,38);(1,3,4,7)(2,6,5,8)(9,13,14,30,16,32,15,31)(10,19,20,28,21,35,33,40)(11,23,24,38,25,39,18,34)(12,26,17,29,22,36,27,37)

Polycyclic presentation:

2^9|b^2=gh,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 379525);