#486088

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	#33692	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 637 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)(6,31)(7,34)(8,14)(9,18)(10,24)(12,22)(13,28)(16,37)(17,49)(19,57)(20,50)(23,44)(25,42)(26,58)(27,40)(29,61)(30,41)(32,47)(33,62)(35,45)(36,55)(38,56)(39,51)(43,60)(46,59)(48,52)(53,64)(54,63);(1,3)(2,9)(4,16)(5,26)(6,32)(7,35)(8,20)(10,36)(11,43)(12,45)(13,47)(14,38)(15,31)(17,29)(18,22)(19,28)(21,58)(23,54)(24,52)(25,60)(27,56)(30,55)(33,51)(34,39)(37,44)(40,57)(41,64)(42,49)(46,63)(48,61)(50,62)(53,59);(1,4)(2,10,14,21)(3,16)(5,27)(6,29)(7,25)(8,24)(9,36,38,58)(11,12,41,40)(13,42,34,46)(15,48,50,37)(17,32)(18,55)(19,53)(20,52)(22,30)(23,33)(26,56)(28,59)(31,61,62,44)(35,60)(39,63,47,49)(43,45,64,57)(51,54);(1,5,28,23,8,30,7,29)(2,11,13,44,14,41,34,61)(3,17,35,55,20,54,19,26)(4,22,59,6,24,27,25,33)(9,37,39,43,38,48,47,64)(10,40,42,62,21,12,46,31)(15,49,45,36,50,63,57,58)(16,51,60,56,52,32,53,18)

Polycyclic presentation:

2^9|c^2=hi,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	#72250797
maximal normal subgroup of	#72250797
maximal quotient of	#72250961

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393284);