#145153

order	512
nilpotency class	4
isoclinism	isoclinic class of #145153
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10431
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28249	C₂
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

324 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,16)(4,22)(5,14)(6,31)(7,33)(8,12)(9,15)(10,18)(11,25)(13,32)(17,41)(19,49)(20,37)(21,39)(23,48)(24,47)(26,42)(27,57)(28,43)(29,40)(30,46)(34,60)(35,45)(36,55)(38,53)(44,59)(50,64)(51,63)(52,62)(54,56)(58,61);(1,3)(2,10)(4,23)(5,29)(6,19)(7,34)(8,18)(9,21)(11,41)(12,39)(13,37)(14,16)(15,40)(17,26)(20,30)(22,56)(24,55)(25,52)(27,54)(28,58)(31,38)(32,53)(33,49)(35,50)(36,43)(42,64)(44,62)(45,63)(46,60)(47,48)(51,59)(57,61);(1,4)(2,11)(3,17,21,51)(5,27)(6,26)(7,35)(8,24)(9,28)(10,36,40,56)(12,44)(13,47)(14,42)(15,45)(16,48,39,61)(18,52,29,50)(19,54,20,55)(22,31)(23,34,58,53)(25,32)(30,59)(33,57)(37,63,49,41)(38,62,60,64)(43,46)

Polycyclic presentation:

2^9|c^2=dg,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fi,g^a=gi,c^b=cf,d^b=di,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fhi,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, 52349);