#143954

order	512
nilpotency class	4
isoclinism	isoclinic class of #143954
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1687
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10260
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,18)(4,24)(5,31)(6,35)(7,14)(8,16)(9,15)(10,17)(11,20)(12,27)(13,34)(19,38)(21,40)(22,57)(23,49)(25,60)(26,44)(28,42)(29,43)(30,48)(32,50)(33,46)(36,39)(37,47)(41,55)(45,61)(51,52)(53,64)(54,63)(56,62)(58,59);(1,3)(2,11)(4,25)(5,32)(6,21)(7,36)(8,37)(9,22)(10,23)(12,41)(13,47)(14,49)(15,50)(16,39)(17,40)(18,31)(19,26)(20,34)(24,59)(27,58)(28,55)(29,56)(30,52)(33,53)(35,57)(38,43)(42,63)(44,60)(45,51)(46,62)(48,64)(54,61);(1,4)(2,12)(3,19)(5,33)(6,26)(7,27)(8,29)(9,28)(10,30)(11,38)(13,48)(14,42)(15,45)(16,44)(17,46)(18,51)(20,55)(21,56)(22,58)(23,54)(24,35)(25,36)(31,43)(32,52)(34,61)(37,53)(39,64)(40,63)(41,47)(49,60)(50,59)(57,62)

Polycyclic presentation:

2^9|d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=dgi,e^c=eh,f^c=fg,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, 51150);