#153253

order	512
nilpotency class	5
isoclinism	isoclinic class of #153253
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13381
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	#33607	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

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

normal subgroups only

This group has 276 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:

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

Polycyclic presentation:

2^9|c^2=dh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=egi,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	#72224988, #72224997, #72224998, …
maximal normal subgroup of	#72224988, #72224997, #72224998, …
maximal quotient of	C₂×#153253

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60449);