#503179

order	512
nilpotency class	4
isoclinism	isoclinic class of #503179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25757
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=e,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410375);