#135662

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄²

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

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,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, 42858);