#509506

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

607 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)(3,5)(4,6)(7,8)(9,10)(11,23)(12,29)(13,19)(14,17)(15,21)(16,26)(18,46)(20,37)(22,36)(24,41)(25,56)(27,60)(28,51)(30,64)(31,43)(32,48)(33,63)(34,50)(35,44)(38,49)(39,62)(40,57)(42,59)(45,67)(47,65)(52,66)(53,55)(54,69)(58,72)(61,70)(68,71);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,30)(13,26)(14,27)(15,24)(17,42)(18,47)(19,23)(20,39)(21,51)(22,41)(25,31)(28,36)(29,56)(32,65)(33,58)(34,54)(35,59)(37,55)(38,68)(40,48)(43,64)(44,60)(45,53)(46,57)(49,70)(50,72)(52,71)(61,66)(62,67)(63,69);(1,4,5,8)(2,6,3,7)(10,17)(11,24)(14,37)(15,36)(16,39)(18,43)(19,44)(20,22)(21,45)(23,53)(25,54)(26,28)(27,59)(29,48)(30,58)(32,38)(33,34)(35,67)(40,69)(41,60)(42,51)(46,66)(47,72)(49,63)(50,52)(56,71)(61,68)(64,70);(2,3)(6,7)(9,12)(10,18)(11,25)(13,31)(14,38)(15,33)(16,40)(17,43)(19,48)(20,50)(21,49)(22,52)(23,47)(24,54)(26,30)(27,61)(28,58)(29,44)(32,37)(34,36)(35,66)(39,69)(41,70)(42,56)(45,63)(46,67)(51,71)(53,72)(55,57)(59,68)(60,64)(62,65)

Polycyclic presentation:

2^9|b^2=ei,c^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fh,h^d=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, 416702);