#151937

order	512
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#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 ◁ C₈ ◁ 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:

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

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,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, 59133);