#145562

order	512
nilpotency class	4
isoclinism	isoclinic class of #144059
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
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

262 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 74 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)(5,6,12,16)(7,18,23,48)(8,24)(9,29,33,46)(10,17,11,35)(13,36,38,20)(14,34)(15,31,44,32)(19,49)(21,43,22,54)(25,57)(26,40)(27,42)(28,41)(30,45,52,39)(37,59)(47,53)(50,65)(51,68)(55,58)(56,60)(61,67)(62,64)(63,66);(5,7,12,23)(6,13,16,38)(8,25,28,19)(9,30,33,52)(10,22,11,21)(14,37,42,66)(15,43,44,54)(17,45,35,39)(18,46,48,29)(20,31,36,32)(24,55,41,65)(26,59,61,63)(27,51,34,53)(40,57,67,49)(47,56,68,62)(50,60,58,64);(2,4)(5,8,9,26,10,34,31,64,12,28,33,61,11,27,32,60)(6,14,15,40,35,24,46,56,16,42,44,67,17,41,29,62)(7,19,20,50,21,53,52,59,23,25,36,58,22,51,30,63)(13,37,18,47,45,65,54,49,38,66,48,68,39,55,43,57)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=d,d^2=gi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fh,g^b=gi,e^c=eghi,f^c=fgh,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, 52758);