#143263

order	512
nilpotency class	4
isoclinism	isoclinic class of #143262
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10048
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

396 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50459);