#263531

order	512
nilpotency class	3
isoclinism	isoclinic class of #263527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

838 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 140 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gh,d^c=dh,f^c=fi,g^c=gh

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, 170727);