#153393

order	512
nilpotency class	5
isoclinism	isoclinic class of #153081
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2218
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#13515
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

163 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 163 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320831, #72320849, #72320895
maximal normal subgroup of	#72320831, #72320849, #72320895

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60589);