#144249

order	512
nilpotency class	4
isoclinism	isoclinic class of #144213
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	C₂⁴	#135
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

392 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 66 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)(9,14)(10,13)(11,15)(12,16)(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)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19,24,36)(18,25,30,51)(20,38,41,64)(21,43,46,66)(22,34,23,35)(26,52,55,77)(27,56,58,67)(28,50,29,59)(31,57,61,42)(32,47,65,40)(33,44,48,45)(37,68,54,49)(39,73,75,63)(53,60,72,70)(62,78,79,69)(71,76,80,74);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(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=i,c^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,e^b=eh,f^b=fi,g^b=gi,e^c=ei,f^c=fi

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, 51445);