#500470

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
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	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

755 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,27)(13,33)(14,35)(15,19)(16,21)(18,30)(20,39)(22,52)(23,47)(24,58)(25,41)(26,43)(28,61)(29,48)(31,65)(32,45)(34,49)(36,67)(37,46)(38,66)(40,55)(42,60)(44,59)(50,57)(51,53)(54,70)(56,69)(62,71)(63,72)(64,68);(2,5)(6,7)(9,11,16,26)(10,17,21,43)(12,28,29,62)(13,25,15,23)(14,36,39,57)(18,44,45,72)(19,47,33,41)(20,50,35,67)(22,53,54,31)(24,34,60,38)(27,61,48,71)(30,59,32,63)(37,55,64,56)(40,68,69,46)(42,66,58,49)(51,70,65,52);(9,12,16,29)(10,18,21,45)(11,22,26,54)(13,32,15,30)(14,37,39,64)(17,40,43,69)(19,48,33,27)(20,51,35,65)(23,56,25,55)(24,59,60,63)(28,57,62,36)(31,34,53,38)(41,70,47,52)(42,71,58,61)(44,67,72,50)(46,66,68,49);(1,3,4,8)(2,6,5,7)(12,29)(18,45)(22,54)(27,48)(28,62)(30,32)(31,53)(37,64)(40,69)(44,72)(46,68)(51,65)(52,70)(55,56)(59,63)(61,71)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=e,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=di,f^c=fi,g^c=gh,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, 407666);