Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6473174 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3455 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(3,7)(6,8)(9,10,16,21,15,22,14,20)(11,23,27,17,28,46,26,39)(12,29,35,41,31,42,32,18)(13,19,36,45,33,43,34,44)(24,49,53,47,51,37,52,65)(25,48,56,38,54,66,55,62)(30,57,60,64,59,63,58,40)(50,68,70,67,71,61,69,72);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,24)(13,25)(14,27)(15,28)(16,26)(18,37)(19,38)(20,46)(21,23)(22,39)(29,47)(30,50)(31,51)(32,53)(33,54)(34,56)(35,52)(36,55)(40,61)(41,49)(42,65)(43,62)(44,66)(45,48)(57,67)(58,70)(59,71)(60,69)(63,72)(64,68);(2,5)(6,8)(9,12,16,35,15,31,14,32)(10,18,20,42,22,41,21,29)(11,24,26,52,28,51,27,53)(13,30,36,60,33,59,34,58)(17,37,46,65,39,49,23,47)(19,40,44,63,43,64,45,57)(25,50,55,69,54,71,56,70)(38,61,66,72,62,68,48,67);(2,5)(6,8)(9,13)(10,19)(11,25)(12,30)(14,34)(15,33)(16,36)(17,38)(18,40)(20,44)(21,45)(22,43)(23,48)(24,50)(26,55)(27,56)(28,54)(29,57)(31,59)(32,58)(35,60)(37,61)(39,62)(41,64)(42,63)(46,66)(47,67)(49,68)(51,71)(52,69)(53,70)(65,72);(1,3,4,7)(2,6,5,8)(12,31)(13,33)(18,41)(19,43)(24,51)(25,54)(29,42)(32,35)(34,36)(37,49)(38,62)(44,45)(47,65)(48,66)(52,53)(55,56)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6401449);