#424240

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1557 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ef,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,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, 331436);