#153419

order	512
nilpotency class	5
isoclinism	isoclinic class of #153419
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13539
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320657, #72321401, #72321450
maximal normal subgroup of	#72320657, #72321401, #72321450

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60615);