#508181

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 ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

851 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)(11,13)(12,23)(14,30)(15,19)(16,22)(17,26)(18,33)(20,36)(21,32)(24,31)(25,37)(27,35)(28,38)(29,39)(34,40);(2,5)(6,8)(9,11,13,10)(12,24,25,34)(14,21,29,18)(15,19,16,22)(17,27,20,28)(23,40,37,31)(26,38,36,35)(30,33,39,32);(3,7)(6,8)(9,12)(10,17)(11,20)(13,25)(14,31)(15,28)(16,27)(18,35)(19,24)(21,38)(22,34)(23,30)(26,33)(29,40)(32,36)(37,39);(1,3,4,7)(2,6,5,8)(12,25)(17,20)(23,37)(24,34)(26,36)(27,28)(31,40)(35,38)

Polycyclic presentation:

2^9|b^2=i,d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,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, 415377);