#135370

order	512
nilpotency class	4
isoclinism	isoclinic class of #124728
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1502
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8716
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

319 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,15,19,34)(4,21,24,53)(5,11)(6,27,29,37)(7,13)(8,14)(9,17,33,20)(10,23)(12,30,39,26)(16,43)(18,32)(22,38,25,56)(28,40)(31,58,59,49)(35,54)(36,55)(41,62,61,45)(42,48)(44,63)(46,52)(47,64)(50,57)(51,60);(1,3)(2,9,14,33)(4,16)(5,17)(6,18)(7,20)(8,19)(10,31,36,59)(11,15,13,34)(12,32,39,60)(21,52,53,63)(22,44)(23,45)(24,47)(25,46)(26,48)(27,57,37,42)(28,49)(29,51)(30,50)(35,41,40,61)(38,43,56,64)(54,58)(55,62);(1,4,5,22,8,24,7,25)(2,10,11,35,14,36,13,40)(3,16,17,44,19,47,20,46)(6,28,26,23,29,54,30,55)(9,31,15,41,33,59,34,61)(12,38,37,53,39,56,27,21)(18,49,48,45,51,58,50,62)(32,43,42,63,60,64,57,52)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=d,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,f^b=fi,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, 42566);