#128483

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8160
prime radical	#532	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=h,h^2=i,b^a=bd,d^a=di,f^a=fg,c^b=ch,d^b=dg,e^b=ei,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343150, #72343184, #72343187
maximal normal subgroup of	#72343150, #72343184, #72343187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35679);