#134460

order	512
nilpotency class	4
isoclinism	isoclinic class of #124838
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1507
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8573
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

382 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fghi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41656);