#151938

order	512
nilpotency class	5
isoclinism	isoclinic class of #13595
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#354
commutator	C₄²	C₂×C₄²
Frattini, prime radical, cosocle	#532	C₂³
socle	C₂	C₄×#354
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

317 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eh,d^2=hi,e^2=i,g^2=i,b^a=bd,c^a=ci,d^a=dh,f^a=fghi,g^a=gi,d^b=dg,e^b=ei,g^b=gh,h^b=hi,f^d=fhi,g^f=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, 59134);