#484990

order	512
nilpotency class	4
isoclinism	isoclinic class of #484989
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 669 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)(3,15,19,37,20,44,18,43)(4,21,25,34,22,36,24,10)(5,26,28,11,29,40,30,39)(9,27,41,17,33,50,32,51)(16,45,49,59,46,60,48,42)(23,35,56,53,54,38,57,61)(31,47,62,64,58,63,52,55);(1,3,8,20)(2,9,14,33)(4,16,22,46)(5,27,29,50)(6,19,7,18)(10,31,34,58)(11,37,39,43)(12,32,13,41)(15,26,44,40)(17,28,51,30)(21,52,36,62)(23,55,54,64)(24,49,25,48)(35,60,38,45)(42,53,59,61)(47,56,63,57);(1,4)(2,10,14,34)(3,16)(5,23)(6,24)(7,25)(8,22)(9,31,33,58)(11,38,39,35)(12,21,13,36)(15,42,44,59)(17,47)(18,48)(19,49)(20,46)(26,53,40,61)(27,55)(28,56)(29,54)(30,57)(32,52,41,62)(37,45,43,60)(50,64)(51,63);(4,22)(10,34)(16,46)(21,36)(23,54)(24,25)(31,58)(35,38)(42,59)(45,60)(47,63)(48,49)(52,62)(53,61)(55,64)(56,57)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,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^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, 392186);