#153306

order	512
nilpotency class	5
isoclinism	isoclinic class of #153306
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13424
commutator, Frattini, cosocle	C₂×C₄⋊Q₈	C₂³
prime radical	C₂×2₋¹⁺⁴	D₄
LCS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

326 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,13,18,8)(4,19)(5,24)(6,29)(7,10)(9,21)(11,27)(12,31)(14,44,46,62)(15,33,17,35)(16,48,25,60)(20,36,49,52)(22,56)(23,38)(26,61)(28,42)(30,39)(32,45,43,51)(34,50,40,47)(37,59)(41,57)(53,55)(54,64,58,63);(1,3)(2,8)(4,20)(5,25)(6,15)(7,18)(9,36)(10,13)(11,40)(12,33)(14,22)(16,28)(17,31)(19,52)(21,49)(23,58)(24,48)(26,46)(27,47)(29,35)(30,50)(32,37)(34,39)(38,64)(41,43)(42,60)(44,61)(45,59)(51,57)(53,63)(54,55)(56,62);(1,4)(2,9,10,19)(3,14)(5,26)(6,23)(7,21)(8,32,13,43)(11,41,39,37)(12,38,29,53)(15,45)(16,49)(17,51)(18,46)(20,25)(22,28)(24,56,42,61)(27,59)(30,57)(31,55)(33,62,35,44)(34,52,40,36)(47,54)(48,63,60,64)(50,58)

Polycyclic presentation:

2^9|a^2=h,c^2=d,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fgh,g^a=gi,c^b=cf,e^b=eg,f^b=fi,g^b=gi,h^b=hi,e^c=ehi,f^c=fi,g^c=gh,f^e=fh,g^f=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60502);