#484659

order	512
nilpotency class	4
isoclinism	isoclinic class of #484658
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#33530	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

869 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 869 conjugacy classes of subgroups — too many to draw. Only its 136 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)(4,20)(5,25)(6,13)(7,12)(8,14)(9,17)(10,23)(11,28)(16,44)(18,51)(19,43)(21,38)(22,37)(24,36)(26,33)(27,60)(29,41)(30,40)(31,39)(32,54)(34,49)(35,50)(42,47)(45,64)(46,52)(48,62)(53,55)(56,58)(57,63)(59,61);(1,3,8,19)(2,9,14,35)(4,16,21,45)(5,26,29,49)(6,18,7,31)(10,32,36,61)(11,39,40,51)(12,34,13,33)(15,41,43,25)(17,30,50,28)(20,52,38,62)(22,54,27,59)(23,47,24,57)(37,44,60,64)(42,53,63,58)(46,56,48,55);(1,4,8,21)(2,10)(3,16,19,45)(5,27,29,22)(6,23,7,24)(9,32)(11,37)(12,20)(13,38)(14,36)(15,42)(17,48,50,46)(18,47,31,57)(25,58)(26,59,49,54)(28,56,30,55)(33,62)(34,52)(35,61)(39,44)(40,60)(41,53)(43,63)(51,64);(4,21)(10,36)(16,45)(20,38)(22,27)(23,24)(32,61)(37,60)(42,63)(44,64)(46,48)(47,57)(52,62)(53,58)(54,59)(55,56)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391855);