#512217

order	512
nilpotency class	4
isoclinism	isoclinic class of #511963
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×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

578 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)(3,6)(4,5)(7,8)(9,10,17,24)(11,19,29,49)(12,18,32,46)(13,33,38,52)(14,22,40,42)(15,21,25,54)(16,41,39,23)(20,37,53,36)(26,57,61,68)(27,48,55,45)(28,62,31,44)(30,63,65,69)(34,50,67,72)(35,66,56,51)(43,58,70,64)(47,60,71,59);(2,5)(6,8)(9,11,25,55,16,28,14,12,17,29,15,27,39,31,40,32)(10,18,42,62,23,45,21,19,24,46,22,44,41,48,54,49)(13,26,56,64,37,60,34,30,38,61,35,58,36,59,67,65)(20,43,66,57,52,69,50,47,53,70,51,68,33,63,72,71);(1,3,4,7)(2,6,5,8)(9,12,14,31,39,55,25,29,17,32,40,28,16,27,15,11)(10,19,21,48,41,62,42,46,24,49,54,45,23,44,22,18)(13,30,34,59,36,64,56,61,38,65,67,60,37,58,35,26)(20,47,50,63,33,57,66,70,53,71,72,69,52,68,51,43);(9,13)(10,20)(11,26)(12,30)(14,34)(15,35)(16,37)(17,38)(18,43)(19,47)(21,50)(22,51)(23,52)(24,53)(25,56)(27,58)(28,60)(29,61)(31,59)(32,65)(33,41)(36,39)(40,67)(42,66)(44,68)(45,69)(46,70)(48,63)(49,71)(54,72)(55,64)(57,62)

Polycyclic presentation:

2^9|a^2=i,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,c^b=ci,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419413);