#136389

order	512
nilpotency class	4
isoclinism	isoclinic class of #136388
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8890
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

323 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dg,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,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, 43585);