#489070

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

689 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=i,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,d^b=di,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fhi,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, 396266);