#135329

order	512
nilpotency class	4
isoclinism	isoclinic class of #124660
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1510
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8713
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

497 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=dg,e^2=h,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^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fgh,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, 42525);