#486089

order	512
nilpotency class	4
isoclinism	isoclinic class of #486086
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18186
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#12170	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

637 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,15)(4,21)(5,11)(6,29)(7,34)(8,14)(9,18)(10,24)(12,22)(13,32)(16,50)(17,48)(19,57)(20,49)(23,59)(25,41)(26,36)(27,40)(28,45)(30,47)(31,44)(33,62)(35,46)(37,53)(38,56)(39,51)(42,54)(43,60)(52,58)(55,63)(61,64);(1,3)(2,9)(4,16)(5,26)(6,30)(7,35)(8,20)(10,36)(11,42)(12,46)(13,47)(14,38)(15,29)(17,23)(18,22)(19,32)(21,58)(24,53)(25,61)(27,56)(28,52)(31,55)(33,51)(34,39)(37,44)(40,57)(41,63)(43,48)(45,64)(49,62)(50,59)(54,60);(1,4,8,24)(2,10)(3,16,20,53)(5,27,28,22)(6,31,33,23)(7,25,32,60)(9,36)(11,12)(13,41)(14,21)(15,37)(17,30,55,51)(18,26,56,52)(19,54,35,61)(29,44)(34,43)(38,58)(39,48)(40,45)(42,46)(47,63)(49,50)(57,64)(59,62);(1,5,7,23,8,28,32,31)(2,11,34,59,14,45,13,44)(3,17,19,26,20,55,35,52)(4,22,25,33,24,27,60,6)(9,37,47,64,38,50,39,42)(10,40,43,29,21,12,41,62)(15,48,57,36,49,63,46,58)(16,51,54,18,53,30,61,56)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250797
maximal normal subgroup of	#72250797
maximal quotient of	#72257006

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393285);