#135791

order	512
nilpotency class	4
isoclinism	isoclinic class of #135790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1846
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 226 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=e,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,f^b=fi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42987);