#486696

order	512
nilpotency class	4
isoclinism	isoclinic class of #486696
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18203
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#62327	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 673 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=g,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=egi,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	#72250541, #72250559, #72250797, …
maximal normal subgroup of	#72250541, #72250559, #72250797, …
maximal quotient of	#72250959

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393892);