#146229

order	512
nilpotency class	4
isoclinism	isoclinic class of #146219
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=de,c^2=i,d^2=g,e^2=g,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,d^c=dgi,e^c=egi,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53425);