#486343

order	512
nilpotency class	4
isoclinism	isoclinic class of #486340
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
prime radical	#24287	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

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

normal subgroups only

This group has 859 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)(3,17)(4,23)(5,28)(6,32)(7,15)(8,14)(9,16)(10,20)(11,26)(12,25)(13,24)(18,46)(19,59)(21,62)(22,53)(27,49)(29,52)(30,44)(31,47)(33,40)(34,50)(35,45)(36,43)(37,48)(38,60)(39,57)(41,54)(42,61)(51,56)(55,63)(58,64);(1,3,9,22)(2,10,16,42)(4,18,27,58)(5,29,31,55)(6,33,34,54)(7,21,8,37)(11,38,45,59)(12,46,44,64)(13,48,43,62)(14,41,15,40)(17,50,53,32)(19,30,60,25)(20,36,61,24)(23,63,49,52)(26,56,35,57)(28,39,47,51);(1,4)(2,11)(3,18)(5,24)(6,30)(7,35)(8,26)(9,27)(10,38)(12,43)(13,44)(14,49)(15,23)(16,45)(17,51)(19,54)(20,29)(21,57)(22,58)(25,34)(28,50)(31,36)(32,47)(33,60)(37,56)(39,53)(40,63)(41,52)(42,59)(46,62)(48,64)(55,61);(1,5,7,30,9,31,8,25)(2,12,14,47,16,44,15,28)(3,19,37,55,22,60,21,29)(4,24,35,6,27,36,26,34)(10,39,40,64,42,51,41,46)(11,43,49,32,45,13,23,50)(17,52,62,38,53,63,48,59)(18,54,56,61,58,33,57,20)

Polycyclic presentation:

2^9|b^2=i,d^2=gi,e^2=gi,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=eg,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	#72307367, #72307417, #72307553, …
maximal normal subgroup of	#72307367, #72307417, #72307553, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393539);