#325413

order	512
nilpotency class	3
isoclinism	isoclinic class of #278051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232609);