#145574

order	512
nilpotency class	4
isoclinism	isoclinic class of #144073
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1675
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 232 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,11,15)(7,17,20,49)(8,22)(9,27,31,47)(10,32,29,16)(12,34,37,19)(13,38)(14,43,45,30)(18,50)(21,54,33,44)(23,57)(24,40)(25,42)(26,41)(28,46,58,36)(35,63)(39,51)(48,52)(53,68)(55,59)(56,60)(61,67)(62,65)(64,66);(5,7,11,20)(6,12,15,37)(8,23)(9,28,31,58)(10,33,29,21)(13,39)(14,44,45,54)(16,46,32,36)(17,47,49,27)(18,38)(19,43,34,30)(22,55)(24,51)(25,53)(26,52)(35,41)(40,57)(42,66)(48,67)(50,62)(56,68)(59,60)(61,64)(63,65);(2,4)(5,8,9,24,29,38,43,65,11,26,31,61,10,25,30,60)(6,13,14,40,32,22,47,56,15,42,45,67,16,41,27,62)(7,18,19,51,33,23,58,59,20,53,34,64,21,52,28,63)(12,35,17,48,46,66,54,68,37,55,49,57,36,39,44,50)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=d,d^2=gi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=gi,e^c=eghi,f^c=fghi,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, 52770);