#135664

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	#1568
prime radical	C₂³	C₂×#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 210 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)(3,6)(4,5)(7,8)(9,10,15,20,16,23,38,52,17,24,14,39,37,22,13,21)(11,25,29,44,30,54,61,45,31,55,28,43,60,42,27,18)(12,32,34,51,35,47,33,49,36,64,40,19,46,48,65,50)(26,41,57,66,58,68,56,63,59,69,62,53,70,67,72,71);(2,5)(6,8)(9,11,13,27,37,60,14,28,17,31,38,61,16,30,15,29)(10,18,39,55,23,44,21,42,24,45,20,25,22,43,52,54)(12,26,33,56,46,70,34,57,36,59,65,72,35,58,40,62)(19,41,48,66,50,68,32,63,51,69,47,53,49,67,64,71);(1,3,4,7)(2,6,5,8)(9,12,13,33,37,46,14,34,17,36,38,65,16,35,15,40)(10,19,20,47,23,50,52,64,24,51,39,48,22,49,21,32)(11,26,27,56,60,70,28,57,31,59,61,72,30,58,29,62)(18,41,25,53,44,68,54,71,45,69,55,66,43,67,42,63)

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,h^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, 42860);