#512214

order	512
nilpotency class	4
isoclinism	isoclinic class of #511960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

587 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)(3,6)(4,5)(7,8)(9,10)(11,25)(12,18)(13,35)(14,41)(15,44)(16,45)(17,24)(19,30)(20,38)(21,26)(22,43)(23,42)(27,59)(28,48)(29,47)(31,64)(32,57)(33,56)(34,49)(36,51)(37,70)(39,54)(40,53)(46,60)(50,62)(52,58)(55,61)(63,67)(65,72)(66,71)(68,69);(2,5)(6,8)(9,11,26,33,16,29,43,34,17,30,15,28,42,32,14,12)(10,18,41,57,23,48,44,19,24,49,22,47,45,56,21,25)(13,27,58,65,39,62,69,66,40,63,37,60,38,61,36,31)(20,46,70,67,53,71,68,50,54,72,52,59,35,64,51,55);(1,3,4,7)(2,6,5,8)(9,12,14,32,42,28,15,30,17,34,43,29,16,33,26,11)(10,19,21,48,45,57,22,18,24,25,44,56,23,47,41,49)(13,31,36,61,38,60,37,63,40,66,69,62,39,65,58,27)(20,50,51,71,35,67,52,46,54,55,68,64,53,59,70,72);(9,13)(10,20)(11,27)(12,31)(14,36)(15,37)(16,39)(17,40)(18,46)(19,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,58)(28,60)(29,62)(30,63)(32,61)(33,65)(34,66)(35,45)(38,42)(41,70)(43,69)(44,68)(47,59)(48,71)(49,72)(56,64)(57,67)

Polycyclic presentation:

2^9|b^2=fi,c^2=e,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419410);