#153319

order	512
nilpotency class	5
isoclinism	isoclinic class of #153319
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13448
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#2263	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 260 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)(3,17)(4,10)(5,28)(6,34)(7,35)(8,37)(9,16)(11,20)(12,24)(13,29)(14,25)(15,36)(18,42)(19,50)(21,58)(22,59)(23,45)(26,47)(27,39)(30,43)(31,62)(32,46)(33,44)(38,56)(40,60)(41,61)(48,54)(49,53)(51,52)(55,64)(57,63);(2,10)(3,18)(4,24)(5,29)(6,27)(7,25)(8,36)(11,40)(12,14)(13,37)(15,47)(16,39)(17,45)(19,55)(20,56)(21,57)(22,30)(23,54)(26,33)(28,35)(31,32)(34,46)(41,58)(42,50)(43,49)(44,62)(48,63)(51,64)(52,53)(60,61);(1,3,4,19,25,52,5,30,8,22,26,53,31,55,6,18,9,23,27,57,32,61,33,20,36,56,29,60,7,21,24,54)(2,11,12,41,35,63,13,45,15,42,44,64,46,49,39,59,16,43,34,51,62,50,47,17,37,48,28,58,14,40,10,38)

Polycyclic presentation:

2^9|c^2=d,d^2=gh,e^2=gi,f^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,f^a=fghi,g^a=gh,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,f^b=fgh,g^b=gh,h^b=hi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320815, #72321401, #72321435
maximal normal subgroup of	#72320815, #72321401, #72321435

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60515);