#500937

order	512
nilpotency class	4
isoclinism	isoclinic class of #499736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408133);