#508060

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1108 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)(12,18)(13,19)(14,17)(15,33)(16,25)(20,30)(21,29)(23,37)(24,34)(26,40)(27,36)(28,35)(31,38)(32,39);(2,5)(6,8)(9,11)(10,16)(12,24)(13,25)(14,32)(15,23)(17,36)(18,35)(19,22)(20,37)(21,34)(26,30)(27,31)(28,29)(33,40)(38,39);(1,3)(2,6)(4,7)(5,8)(10,17)(11,23)(12,29)(14,30)(15,31)(16,34)(18,33)(19,38)(20,21)(22,35)(24,27)(25,26)(28,32)(37,40);(3,7)(6,8)(9,12,13,29)(10,18,19,21)(11,24,25,28)(14,30,31,15)(16,35,22,34)(17,20,38,33)(23,32,26,27)(36,37,39,40)

Polycyclic presentation:

2^9|d^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,d^c=de,f^c=fh,g^c=ghi,h^c=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, 415256);