#512069

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

959 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)(9,10,16,23)(11,17,28,44)(12,29,32,51)(13,33,37,50)(14,22,15,21)(18,45,48,30)(19,38,49,31)(20,35,46,36)(24,53,56,70)(25,57,61,69)(26,43,27,42)(34,52,63,47)(39,64,67,54)(40,62,68,55)(41,59,65,60)(58,71,72,66);(2,5)(6,8)(9,11,16,28)(10,17,23,44)(12,24,32,56)(13,25,37,61)(14,26,15,27)(18,39,48,67)(19,40,49,68)(20,41,46,65)(21,42,22,43)(29,53,51,70)(30,54,45,64)(31,55,38,62)(33,57,50,69)(34,58,63,72)(35,59,36,60)(47,66,52,71);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,24)(13,34)(14,31)(15,38)(16,32)(17,39)(19,46)(20,49)(21,47)(22,52)(23,48)(25,58)(26,55)(27,62)(28,56)(29,33)(30,36)(35,45)(37,63)(40,65)(41,68)(42,66)(43,71)(44,67)(50,51)(53,57)(54,60)(59,64)(61,72)(69,70);(9,12)(10,19)(11,24)(13,30)(14,38)(15,31)(16,32)(17,40)(18,46)(20,48)(21,51)(22,29)(23,49)(25,54)(26,62)(27,55)(28,56)(33,52)(34,36)(35,63)(37,45)(39,65)(41,67)(42,70)(43,53)(44,68)(47,50)(57,71)(58,60)(59,72)(61,64)(66,69)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419265);