#135423

order	512
nilpotency class	4
isoclinism	isoclinic class of #124661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1545
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8724
prime radical	C₂⁴	#101
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

334 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

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

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, 42619);