#501937

order	512
nilpotency class	4
isoclinism	isoclinic class of #500737
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24967
prime radical	#28249	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 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)(3,15)(4,21)(5,28)(6,13)(7,32)(8,14)(9,18)(10,25)(11,29)(12,31)(16,49)(17,53)(19,39)(20,37)(22,40)(23,41)(24,44)(26,45)(27,42)(30,46)(33,38)(34,59)(35,52)(36,56)(43,61)(47,51)(48,55)(50,64)(54,58)(57,63)(60,62);(1,3,8,20)(2,9)(4,22,27,60)(5,17,30,56)(6,18,31,19)(7,33,29,55)(10,40)(11,36)(12,15)(13,37)(14,39)(16,23,52,26)(21,58)(24,54,25,50)(28,48)(32,53)(34,57,61,51)(35,43)(38,46)(41,63)(42,64)(44,62)(45,47)(49,59);(1,4)(2,10)(3,16)(5,26)(6,24)(7,34)(8,27)(9,35)(11,45)(12,42)(13,21)(14,44)(15,47)(17,54)(18,51)(19,57)(20,52)(22,33)(23,30)(25,31)(28,59)(29,61)(32,41)(36,40)(37,63)(38,58)(39,49)(43,46)(48,64)(50,56)(53,62)(55,60);(1,5,6,29,8,30,31,7)(2,11,12,28,14,32,13,46)(3,17,18,55,20,56,19,33)(4,23,24,34,27,26,25,61)(9,36,15,48,39,53,37,38)(10,41,42,43,44,45,21,59)(16,50,51,22,52,54,57,60)(35,62,47,58,49,40,63,64)

Polycyclic presentation:

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

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, 409133);