#484234

order	512
nilpotency class	4
isoclinism	isoclinic class of #484233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1069 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 1069 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)(5,24)(6,13)(7,12)(8,14)(9,17)(10,23)(11,27)(16,44)(18,52)(19,43)(21,40)(22,55)(25,33)(26,37)(28,41)(29,36)(30,39)(31,38)(32,59)(34,50)(35,51)(42,47)(45,62)(46,53)(48,64)(49,61)(54,58)(56,60)(57,63);(1,3,8,19)(2,9,14,35)(4,16,21,45)(5,25,28,50)(6,18,7,31)(10,32,36,60)(11,38,39,52)(12,34,13,33)(15,41,43,24)(17,30,51,27)(20,53,40,61)(22,56,26,59)(23,47,29,48)(37,62,55,44)(42,63,64,54)(46,57,49,58);(1,4)(2,10)(3,16)(5,26)(6,29)(7,23)(8,21)(9,32)(11,37)(12,40)(13,20)(14,36)(15,42)(17,49)(18,48)(19,45)(22,28)(24,54)(25,59)(27,57)(30,58)(31,47)(33,53)(34,61)(35,60)(38,62)(39,55)(41,63)(43,64)(44,52)(46,51)(50,56);(4,21)(10,36)(16,45)(20,40)(22,26)(23,29)(32,60)(37,55)(42,64)(44,62)(46,49)(47,48)(53,61)(54,63)(56,59)(57,58)

Polycyclic presentation:

2^9|b^2=i,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^c=fi,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, 391430);