#135663

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1835
prime radical	C₂³	#427
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

216 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,3,5,4,6,7,8)(9,10)(11,23)(12,30,34,61)(13,36)(14,19)(15,39)(16,22)(17,26)(18,33,46,67)(20,24)(21,37)(25,44,31,40)(27,50)(28,41)(29,52)(32,65,54,47)(35,68,66,45)(38,48)(42,59)(43,53)(49,56,71,63)(51,58)(55,69,62,60)(57,70,64,72);(9,11,24,53,37,58,14,27,16,29,13,26,15,28,38,59)(10,17,19,23,39,50,20,41,22,43,48,52,21,42,36,51)(12,31,54,55,66,57,33,63,34,25,32,62,35,64,67,56)(18,44,45,71,65,70,30,60,46,40,68,49,47,72,61,69);(9,12,13,32,37,66,38,67,16,34,24,54,15,35,14,33)(10,18,19,45,39,65,20,30,22,46,48,68,21,47,36,61)(11,25,26,55,58,64,59,63,29,31,53,62,28,57,27,56)(17,40,23,49,50,72,41,69,43,44,52,71,42,70,51,60)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42859);