#153371

order	512
nilpotency class	5
isoclinism	isoclinic class of #153371
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13492
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

180 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=d,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=di,e^b=ei,e^c=eg,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320657, #72321417, #72321458
maximal normal subgroup of	#72320657, #72321417, #72321458

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60567);