#486079

order	512
nilpotency class	4
isoclinism	isoclinic class of #486077
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18178
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 742 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)(3,14)(4,10)(5,11)(6,25)(7,30)(8,13)(9,17)(12,27)(15,43)(16,44)(18,45)(19,36)(20,42)(21,53)(22,56)(23,57)(24,39)(26,32)(28,58)(29,59)(31,40)(33,49)(34,50)(35,48)(37,54)(38,52)(41,51)(46,63)(47,62)(55,64)(60,61);(1,3)(2,9)(4,15)(5,21)(6,26)(7,31)(8,18)(10,32)(11,37)(12,40)(13,34)(14,25)(16,23)(17,27)(19,47)(20,50)(22,49)(24,55)(28,46)(29,48)(30,35)(33,36)(38,44)(39,61)(41,62)(42,43)(45,59)(51,60)(52,54)(53,56)(57,63)(58,64);(1,4,8,7)(2,10,13,30)(3,15,18,31)(5,19,24,23)(6,27,29,20)(9,32,34,35)(11,36,39,57)(12,25,42,59)(14,43,45,40)(16,21,47,55)(17,48,50,26)(22,52,28,51)(33,61,63,37)(38,56,41,58)(44,53,62,64)(46,60,49,54);(1,5,4,19,8,24,7,23)(2,11,10,36,13,39,30,57)(3,16,31,55,18,47,15,21)(6,28,20,52,29,22,27,51)(9,33,35,37,34,63,32,61)(12,41,25,58,42,38,59,56)(14,44,40,64,45,62,43,53)(17,49,48,54,50,46,26,60)

Polycyclic presentation:

2^9|c^2=i,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,e^c=eh,e^d=ei,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250033, #72307468
maximal normal subgroup of	#72250033, #72307468
maximal quotient of	#72250375, #72280005

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393275);