#505233

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 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)(11,26)(12,32)(13,38)(14,43)(15,45)(16,23)(17,25)(18,28)(19,37)(20,39)(21,33)(22,34)(24,44)(27,50)(29,48)(30,57)(31,47)(35,51)(36,49)(40,54)(41,52)(42,69)(46,61)(53,64)(55,58)(56,59)(60,67)(62,72)(63,65)(66,71)(68,70);(2,5)(6,8)(9,11,15,29,16,31,14,28,17,12,33,37,44,30,34,36)(10,18,22,47,24,48,21,26,25,49,43,57,23,19,45,32)(13,27,40,60,41,62,39,59,42,35,64,66,70,61,58,65)(20,46,52,71,54,51,38,56,55,72,68,67,53,50,69,63);(1,3,4,7)(2,6,5,8)(9,12,34,28,44,31,33,29,17,11,14,36,16,30,15,37)(10,19,45,57,23,49,43,26,25,48,21,47,24,18,22,32)(13,27,58,65,70,61,64,66,42,35,39,59,41,62,40,60)(20,51,69,71,53,46,68,63,55,50,38,67,54,72,52,56);(2,5)(6,8)(9,13,33,64,16,41,34,58,17,42,15,40,44,70,14,39)(10,20,43,68,24,54,45,69,25,55,22,52,23,53,21,38)(11,27,37,66,31,62,36,65,12,35,29,60,30,61,28,59)(18,46,57,67,48,51,32,63,49,72,47,71,19,50,26,56)

Polycyclic presentation:

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

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, 412429);