#143694

order	512
nilpotency class	4
isoclinism	isoclinic class of #143694
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10228
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
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

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

normal subgroups only

This group has 441 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)(4,22)(5,13)(6,12)(7,16)(8,15)(9,14)(10,19)(11,24)(18,33)(20,55)(21,47)(23,38)(25,59)(26,37)(27,44)(28,41)(29,42)(30,45)(31,40)(32,43)(34,54)(35,60)(36,52)(39,58)(46,49)(48,56)(50,64)(51,63)(53,62)(57,61);(1,3)(2,10)(4,18)(5,27)(6,19)(7,31)(8,20)(9,21)(11,36)(12,40)(13,17)(14,34)(15,44)(16,35)(22,56)(23,57)(24,49)(25,51)(26,52)(28,54)(29,60)(30,53)(32,50)(33,38)(37,62)(39,63)(41,55)(42,47)(43,61)(45,64)(46,59)(48,58);(1,4)(2,11)(3,18)(5,24)(6,30)(7,25)(8,32)(9,26)(10,33)(12,22)(13,43)(14,38)(15,39)(16,45)(17,46)(19,53)(20,50)(21,52)(23,29)(27,49)(28,58)(31,51)(34,36)(35,63)(37,41)(40,62)(42,59)(44,64)(47,61)(48,54)(55,56)(57,60)

Polycyclic presentation:

2^9|d^2=gi,e^2=hi,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=dg,g^b=gi,d^c=di,e^c=eh,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, 50890);