#486340

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
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 871 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)(4,24)(5,29)(6,33)(7,15)(8,14)(9,16)(10,20)(11,27)(12,31)(13,25)(18,55)(19,38)(21,48)(22,53)(23,54)(26,47)(28,45)(30,52)(32,46)(34,50)(35,49)(36,43)(37,42)(39,62)(40,56)(41,61)(44,58)(51,57)(59,63)(60,64);(1,3)(2,10)(4,18)(5,30)(6,34)(7,22)(8,21)(9,23)(11,38)(12,44)(13,48)(14,40)(15,50)(16,41)(17,33)(19,31)(20,25)(24,52)(26,60)(27,62)(28,58)(29,39)(32,59)(35,56)(36,57)(37,61)(42,53)(43,64)(45,63)(46,51)(47,55)(49,54);(1,4)(2,11)(3,18)(5,25)(6,26)(7,36)(8,27)(9,28)(10,38)(12,42)(13,47)(14,45)(15,24)(16,43)(17,51)(19,56)(20,30)(21,62)(22,57)(23,58)(29,49)(31,35)(32,37)(33,46)(34,60)(39,54)(40,63)(41,64)(44,53)(48,55)(50,52)(59,61);(1,5,8,31,9,32,7,26)(2,12,15,29,16,47,14,46)(3,19,22,30,23,60,21,59)(4,25,27,35,28,37,36,6)(10,39,40,44,41,51,50,55)(11,42,24,49,43,13,45,33)(17,52,48,38,54,63,53,64)(18,56,57,20,58,34,62,61)

Polycyclic presentation:

2^9|d^2=g,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	#72250520, #72250544, #72250797, …
maximal normal subgroup of	#72250520, #72250544, #72250797, …
maximal quotient of	#72250968

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393536);