#486328

order	512
nilpotency class	4
isoclinism	isoclinic class of #486327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

874 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393524);