#153403

order	512
nilpotency class	5
isoclinism	isoclinic class of #153108
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2246
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13526
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

208 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=deg,d^2=gh,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dg,e^b=eg,g^b=gh,h^b=hi,d^c=dgi,e^c=egi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321265, #72321384, #72321417
maximal normal subgroup of	#72321265, #72321384, #72321417

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60599);