#506407

order	512
nilpotency class	4
isoclinism	isoclinic class of #500592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28251
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|b^2=eg,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,g^c=gi,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, 413603);