#144056

order	512
nilpotency class	4
isoclinism	isoclinic class of #144056
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1697
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10267
prime radical	#33594	C₂
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

345 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=hi,f^2=g,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=dgi,f^b=fgi,g^b=gi,d^c=dg,e^c=eh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51252);