#134422

order	512
nilpotency class	4
isoclinism	isoclinic class of #124974
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1517
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8571
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

405 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,15)(4,21,24,53)(5,13)(6,27,29,37)(7,11)(8,14)(9,17)(10,23)(12,26,39,30)(16,43,46,62)(18,32,50,59)(19,34)(20,33)(22,54,25,38)(28,40)(31,49)(35,55)(36,56)(41,45)(42,48,57,51)(44,52,47,61)(58,63)(60,64);(1,3,5,17,8,20,7,19)(2,9,11,15,14,34,13,33)(4,16,22,44,24,46,25,47)(6,18,26,48,29,50,30,51)(10,31,35,41,36,58,40,60)(12,32,37,42,39,59,27,57)(21,52,38,43,53,61,54,62)(23,45,28,49,56,64,55,63);(1,4)(2,10)(3,16)(5,22)(6,28)(7,25)(8,24)(9,31)(11,35)(12,38)(13,40)(14,36)(15,41)(17,44)(18,49)(19,47)(20,46)(21,27)(23,30)(26,56)(29,55)(32,43)(33,60)(34,58)(37,53)(39,54)(42,61)(45,51)(48,64)(50,63)(52,57)(59,62)

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,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, 41618);