#136190

order	512
nilpotency class	4
isoclinism	isoclinic class of #12173
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1598
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8869
prime radical	#3054	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

354 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=gi,e^2=gi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=ehi,g^b=gi,h^b=hi,d^c=dh,e^c=egi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43386);