#502342

order	512
nilpotency class	4
isoclinism	isoclinic class of #502133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,7)(4,5)(6,8)(9,10,14,20,37,51,36,50,13,19,35,49,16,22,15,21)(11,23,26,55,61,44,27,52,25,17,40,42,28,56,38,43)(12,29,32,53,39,68,67,46,31,65,66,48,34,47,33,18)(24,57,59,70,64,71,30,41,58,69,63,72,60,45,62,54);(2,5)(6,7)(9,11,14,26,37,61,36,27,13,25,35,40,16,28,15,38)(10,17,20,42,51,56,50,43,19,23,49,55,22,44,21,52)(12,30,32,58,39,63,67,60,31,62,66,24,34,59,33,64)(18,45,29,54,53,57,68,70,46,71,65,41,48,69,47,72);(9,12,13,31)(10,18,19,46)(11,24,25,58)(14,33,35,67)(15,32,36,66)(16,39,37,34)(17,41,23,54)(20,47,49,68)(21,29,50,65)(22,53,51,48)(26,62,40,30)(27,63,38,59)(28,64,61,60)(42,71,55,45)(43,57,52,69)(44,72,56,70);(1,3,4,8)(2,6,5,7)(10,19)(17,23)(18,46)(20,49)(21,50)(22,51)(29,65)(41,54)(42,55)(43,52)(44,56)(45,71)(47,68)(48,53)(57,69)(70,72)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi,h^c=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, 409538);