#143174

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
prime radical	#33692	C₂
LCS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 298 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 50370);