#219661

order	512
nilpotency class	3
isoclinism	isoclinic class of #219576
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#540
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

459 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 158 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 — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=e,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dh,f^b=fh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126857);