#143173

order	512
nilpotency class	4
isoclinism	isoclinic class of #143173
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10026
commutator, Frattini, cosocle	#405	C₂³
LCS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 52 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)(3,15)(4,22)(5,10)(6,30)(7,33)(8,14)(9,17)(11,24)(12,29)(13,25)(16,48)(18,47)(19,46)(20,41)(21,49)(23,37)(26,42)(27,57)(28,61)(31,40)(32,43)(34,52)(35,58)(36,54)(38,55)(39,59)(44,60)(45,53)(50,62)(51,64)(56,63);(1,3)(2,9)(4,23)(5,28)(6,16)(7,20)(8,21)(10,35)(11,41)(12,45)(13,37)(14,38)(15,22)(17,24)(18,31)(19,25)(26,52)(27,54)(29,51)(30,62)(32,53)(33,36)(34,40)(39,64)(42,61)(43,48)(44,46)(47,56)(49,57)(50,63)(55,60)(58,59);(2,10)(3,16)(4,24)(5,29)(6,31)(7,25)(11,42)(12,13)(14,30)(15,46)(17,52)(18,19)(20,51)(21,28)(22,39)(23,58)(26,32)(27,60)(33,40)(34,62)(35,45)(36,37)(41,49)(43,44)(47,64)(50,54)(53,55)(56,57)

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=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fg,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50369);