#486087

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	#575	C₂³
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 627 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)(7,33)(8,14)(9,18)(10,24)(12,22)(13,31)(16,48)(17,49)(19,50)(20,58)(23,60)(25,40)(26,35)(27,39)(28,44)(30,46)(32,47)(34,45)(36,54)(37,51)(38,57)(41,53)(42,61)(43,55)(52,59)(56,63)(62,64);(1,3,8,19)(2,9,14,38)(4,16,24,54)(5,26,28,52)(6,30,32,51)(7,34,31,20)(10,35,21,59)(11,41,44,64)(12,45,39,58)(13,46,33,37)(15,47,50,29)(17,55,56,23)(18,27,57,22)(25,62,61,53)(36,60,48,43)(40,63,42,49);(1,4)(2,10,14,21)(3,16)(5,27)(6,23)(7,25)(8,24)(9,35,38,59)(11,12,44,39)(13,40,33,42)(15,48,50,36)(17,30)(18,52)(19,54)(20,53)(22,28)(26,57)(29,60,47,43)(31,61)(32,55)(34,62)(37,49,46,63)(41,45,64,58)(51,56);(1,5,7,23,8,28,31,55)(2,11,33,60,14,44,13,43)(3,17,20,26,19,56,34,52)(4,22,25,32,24,27,61,6)(9,36,46,64,38,48,37,41)(10,39,42,29,21,12,40,47)(15,49,58,35,50,63,45,59)(16,51,53,18,54,30,62,57)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=gh,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	#72307849
maximal normal subgroup of	#72307849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393283);