#142359

order	512
nilpotency class	4
isoclinism	isoclinic class of #142246
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

437 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18,24,30)(19,31,36,61)(20,37,41,54)(21,42,46,57)(22,29,23,28)(25,48,51,33)(26,47,55,40)(27,44,58,45)(32,62,65,79)(34,67,35,56)(38,70,64,60)(39,72,75,53)(43,59,66,50)(49,71,68,80)(52,73,77,63)(69,76,78,74);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19)(18,25)(20,38)(21,43)(22,34)(23,35)(24,36)(26,52)(27,56)(28,50)(29,59)(30,51)(31,42)(32,40)(33,45)(37,68)(39,73)(41,64)(44,48)(46,66)(47,65)(49,54)(53,60)(55,77)(57,61)(58,67)(62,69)(63,75)(70,72)(71,74)(76,80)(78,79);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20)(18,26)(19,32)(21,39)(22,47)(23,40)(24,41)(25,49)(27,53)(28,54)(29,37)(30,55)(31,60)(33,63)(34,38)(35,64)(36,65)(42,69)(43,71)(44,76)(45,74)(46,75)(48,73)(50,77)(51,68)(52,59)(56,79)(57,78)(58,72)(61,70)(62,67)(66,80)

Polycyclic presentation:

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

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, 49555);