Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219525 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,21,25,48)(12,17)(13,18)(14,19)(15,20)(16,24,38,34)(22,50,53,63)(23,43,55,69)(26,40)(27,41)(28,42)(29,59,60,37)(30,44)(31,45)(32,46)(33,47)(35,49)(36,56,39,52)(51,66,71,64)(54,62,72,65)(57,67)(58,68)(61,70);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,22)(13,29)(14,34)(15,25)(17,39)(18,43)(19,48)(20,38)(21,45)(23,32)(24,31)(26,54)(27,56)(28,53)(30,51)(33,60)(35,52)(36,42)(37,46)(40,66)(41,63)(44,65)(47,69)(49,50)(55,57)(58,72)(59,67)(61,71)(62,70)(64,68);(2,6)(5,8)(9,12,15,28)(10,17,20,42)(11,22,25,53)(13,30,33,61)(14,35,31,27)(16,36,38,39)(18,44,47,70)(19,49,45,41)(21,50,48,63)(23,54,55,72)(24,56,34,52)(26,57,58,32)(29,51,60,71)(37,64,59,66)(40,67,68,46)(43,62,69,65);(3,7)(5,8)(9,13,31,57,15,33,14,32)(10,18,45,67,20,47,19,46)(11,23,34,60,25,55,24,29)(12,26,35,61,28,58,27,30)(16,37,48,69,38,59,21,43)(17,40,49,70,42,68,41,44)(22,51,56,72,53,71,52,54)(36,62,50,66,39,65,63,64)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=ei,e^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298475);