#143797

order	512
nilpotency class	4
isoclinism	isoclinic class of #143797
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10235
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28245	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

382 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,17,21,47)(4,22,26,39)(5,27)(6,31)(7,16)(8,15)(9,14)(10,19,36,56)(11,24,40,23)(12,29)(13,30)(18,35)(20,42,33,48)(25,43,34,58)(28,44,55,37)(32,45,60,41)(38,53)(46,59)(49,62)(50,61)(51,64)(52,63)(54,57);(1,3)(2,10)(4,18)(5,28)(6,19)(7,33)(8,20)(9,21)(11,38)(12,42)(13,17)(14,36)(15,44)(16,37)(22,57)(23,59)(24,50)(25,52)(26,53)(27,47)(29,56)(30,55)(31,48)(32,54)(34,51)(35,40)(39,62)(41,63)(43,61)(45,64)(46,58)(49,60);(1,4)(2,11)(3,18)(5,24)(6,32)(7,25)(8,34)(9,26)(10,35)(12,22)(13,43)(14,40)(15,41)(16,45)(17,46)(19,54)(20,51)(21,53)(23,30)(27,58)(28,50)(29,60)(31,39)(33,52)(36,38)(37,63)(42,62)(44,64)(47,61)(48,57)(49,56)(55,59)

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,g^b=gi,d^c=di,e^c=ehi,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, 50993);