#503825

order	512
nilpotency class	4
isoclinism	isoclinic class of #500737
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#26111
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

399 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=efh,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 411021);