#500590

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
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	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

785 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)(3,7)(4,5)(6,8)(9,10)(11,23)(12,28)(13,19)(14,35)(15,22)(16,21)(17,25)(18,31)(20,39)(24,54)(26,57)(27,43)(29,64)(30,62)(32,50)(33,63)(34,51)(36,68)(37,46)(38,48)(40,52)(41,70)(42,69)(44,58)(45,65)(47,67)(49,59)(53,56)(55,71)(60,61)(66,72);(2,5)(6,7)(9,11)(10,17)(12,29)(13,25)(14,36)(15,40)(16,27)(18,44)(19,23)(20,49)(21,52)(22,43)(24,32)(26,34)(28,61)(30,60)(31,66)(33,55)(35,57)(37,56)(38,59)(39,69)(41,46)(42,48)(45,72)(47,53)(50,71)(51,68)(54,63)(58,65)(62,64)(67,70);(9,12,13,30)(10,18,19,45)(11,24,25,55)(14,37,34,67)(15,33,16,32)(17,41,23,53)(20,50,48,63)(21,47,22,46)(26,58,36,66)(27,60,40,29)(28,51,62,35)(31,38,65,39)(42,64,49,61)(43,72,52,44)(54,68,71,57)(56,69,70,59);(1,3,4,8)(2,6,5,7)(10,19)(17,23)(18,45)(20,48)(21,22)(28,62)(35,51)(41,53)(42,49)(43,52)(44,72)(46,47)(50,63)(54,71)(57,68)(61,64)

Polycyclic presentation:

2^9|c^2=i,d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fh,g^c=gi,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, 407786);