#134423

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

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

normal subgroups only

This group has 397 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=hi,f^2=i,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, 41619);