#486344

order	512
nilpotency class	4
isoclinism	isoclinic class of #486340
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18194
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

871 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 871 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=gi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,g^b=gi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250527, #72250549, #72250797, …
maximal normal subgroup of	#72250527, #72250549, #72250797, …
maximal quotient of	#72257008

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393540);