#486337

order	512
nilpotency class	4
isoclinism	isoclinic class of #486333
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

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

normal subgroups only

This group has 880 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)(3,14)(4,10)(5,21)(6,26)(7,30)(8,13)(9,17)(11,24)(12,28)(15,41)(16,44)(18,43)(19,38)(20,40)(22,42)(23,54)(25,51)(27,32)(29,58)(31,39)(33,52)(34,48)(35,47)(36,45)(37,57)(46,62)(49,64)(50,56)(53,61)(55,59)(60,63);(1,3)(2,9)(4,15)(5,22)(6,27)(7,31)(8,18)(10,32)(11,36)(12,39)(13,34)(14,26)(16,24)(17,28)(19,46)(20,48)(21,33)(23,55)(25,53)(29,47)(30,35)(37,61)(38,59)(40,41)(42,54)(43,58)(44,50)(45,57)(49,63)(51,60)(52,56)(62,64);(1,4,8,7)(2,10,13,30)(3,15,18,31)(5,19,25,24)(6,28,29,20)(9,32,34,35)(11,21,38,51)(12,26,40,58)(14,41,43,39)(16,22,46,53)(17,47,48,27)(23,56,57,49)(33,59,60,36)(37,50,54,64)(42,62,61,44)(45,63,55,52);(1,5,4,19,8,25,7,24)(2,11,30,51,13,38,10,21)(3,16,31,53,18,46,15,22)(6,23,20,49,29,57,28,56)(9,33,32,59,34,60,35,36)(12,37,58,64,40,54,26,50)(14,42,41,62,43,61,39,44)(17,45,47,63,48,55,27,52)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250025, #72250033, #72250217, …
maximal normal subgroup of	#72250025, #72250033, #72250217, …
maximal quotient of	#72250318, #72280056

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393533);