#142287

order	512
nilpotency class	4
isoclinism	isoclinic class of #142248
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

558 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 72 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)(9,10,16,21)(11,22,27,52)(12,28,31,18)(13,34,38,50)(14,20,15,39)(17,40,43,24)(19,36,48,37)(23,53,55,51)(25,47,26,58)(29,62,56,72)(30,64,65,68)(32,61,33,46)(35,49,57,42)(41,54,71,70)(44,63,59,69)(45,67,60,66);(5,8)(9,11,16,27)(10,17,21,43)(12,29,31,56)(13,35,38,57)(14,26,15,25)(18,44,28,59)(19,47,48,58)(20,49,39,42)(22,50,52,34)(23,33,55,32)(24,36,40,37)(30,54,65,70)(41,61,71,46)(45,53,60,51)(62,68,72,64)(63,66,69,67);(1,3)(2,5)(4,7)(6,8)(9,12,15,33,16,31,14,32)(10,18,20,46,21,28,39,61)(11,23,26,56,27,55,25,29)(13,30,37,67,38,65,36,66)(17,41,42,44,43,71,49,59)(19,45,34,68,48,60,50,64)(22,51,47,72,52,53,58,62)(24,54,35,69,40,70,57,63)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=dh,e^c=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49483);