#397032

order	512
nilpotency class	3
isoclinism	isoclinic class of #395252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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)(3,7)(4,8)(5,6)(9,10)(11,21)(12,14)(13,29)(15,19)(16,20)(17,24)(18,32)(22,25)(23,37)(26,27)(28,31)(30,34)(33,39)(35,40)(36,38);(2,6)(7,8)(9,11,15,25,16,26,14,24)(10,17,12,27,20,22,19,21)(13,23,31,39,32,40,30,38)(18,33,28,37,29,36,34,35);(1,3,5,4)(2,7,6,8)(9,12,14,10,16,19,15,20)(11,22,24,27,26,17,25,21)(13,28,30,18,32,34,31,29)(23,36,38,37,40,33,39,35);(1,4,5,3)(2,8,6,7)(9,13,14,30,16,32,15,31)(10,18,19,34,20,29,12,28)(11,23,24,38,26,40,25,39)(17,33,21,35,22,36,27,37)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^d=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, 304228);