#486336

order	512
nilpotency class	4
isoclinism	isoclinic class of #486333
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	#24285	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

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

normal subgroups only

This group has 869 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)(3,13)(4,18)(5,24)(6,28)(7,12)(8,15)(9,22)(10,27)(11,20)(14,42)(16,48)(17,41)(19,40)(21,53)(23,50)(25,38)(26,39)(29,36)(30,51)(31,47)(32,57)(33,44)(34,55)(35,43)(37,46)(45,62)(49,56)(52,60)(54,59)(58,64)(61,63);(1,3,7,17)(2,8,12,31)(4,19,23,52)(5,25,26,57)(6,29,20,16)(9,33,35,59)(10,36,37,48)(11,38,28,32)(13,39,41,24)(14,43,45,22)(15,46,47,27)(18,30,50,58)(21,54,55,44)(34,60,53,40)(42,61,62,49)(51,63,64,56);(5,26)(15,47)(21,55)(25,57)(27,46)(44,54)(51,64)(56,63);(1,4,20,43,7,23,6,22)(2,9,28,50,12,35,11,18)(3,14,29,52,17,45,16,19)(5,21,46,63,26,55,27,56)(8,30,38,59,31,58,32,33)(10,34,39,61,37,53,24,49)(13,40,48,62,41,60,36,42)(15,44,57,64,47,54,25,51)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281192, #72281615, #72281659, …
maximal normal subgroup of	#72281192, #72281615, #72281659, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393532);