#464956

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1109 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)(12,26)(13,29)(14,20)(15,23)(16,21)(17,18)(19,33)(24,30)(25,28)(27,35)(31,37)(32,36)(34,39)(38,40);(2,5)(6,8)(9,11,16,26,14,18,15,25)(10,17,21,28,20,22,23,12)(13,24,33,40,31,35,32,39)(19,34,37,30,36,38,29,27);(1,3)(2,6)(4,7)(5,8)(9,12,15,17,14,28,16,22)(10,18,21,26,20,11,23,25)(13,30,32,38,31,27,33,34)(19,24,37,39,36,35,29,40);(3,7)(6,8)(9,13)(10,19)(11,24)(12,27)(14,31)(15,32)(16,33)(17,34)(18,35)(20,36)(21,37)(22,38)(23,29)(25,39)(26,40)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372152);