#153248

order	512
nilpotency class	5
isoclinism	isoclinic class of #153248
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13380
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33617	C₂
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

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

Polycyclic presentation:

2^9|c^2=d,d^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=di,f^b=fi,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	#72320815, #72321417, #72321450
maximal normal subgroup of	#72320815, #72321417, #72321450

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60444);