#140213

order	512
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9560
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

242 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cgh,e^b=ei,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320560, #72320710, #72320780
maximal normal subgroup of	#72320560, #72320710, #72320780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47409);