#484577

order	512
nilpotency class	4
isoclinism	isoclinic class of #484576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60773	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

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

normal subgroups only

This group has 805 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,e^2=g,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=egi,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, 391773);