#153097

order	512
nilpotency class	5
isoclinism	isoclinic class of #153097
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13304
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
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

354 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320815, #72321417, #72321425
maximal normal subgroup of	#72320815, #72321417, #72321425

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60293);