Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1615 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,12,18)(7,19,24,38)(8,25,30,42)(9,31,35,15)(10,17,11,16)(13,23,39,22)(14,28,43,29)(20,36,49,52)(21,44,51,61)(26,54,48,40)(27,56,60,53)(32,37,50,64)(33,45,34,46)(41,59,62,58)(47,63,57,66)(55,68,67,65);(5,7)(6,13)(8,26)(9,32)(10,23)(11,22)(12,24)(14,40)(15,44)(16,38)(17,19)(18,39)(20,28)(21,33)(25,52)(27,57)(29,49)(30,48)(31,61)(34,51)(35,50)(36,42)(37,45)(41,66)(43,54)(46,64)(47,60)(53,68)(55,58)(56,65)(59,67)(62,63);(2,4)(5,8)(6,14)(7,20)(9,27)(10,28)(11,29)(12,30)(13,36)(15,41)(16,42)(17,25)(18,43)(19,40)(21,47)(22,26)(23,48)(24,49)(31,62)(32,55)(33,58)(34,59)(35,60)(37,63)(38,54)(39,52)(44,65)(45,56)(46,53)(50,67)(51,57)(61,68)(64,66);(5,9,10,33,12,35,11,34)(6,15,16,45,18,31,17,46)(7,21,22,32,24,51,23,50)(8,27,28,58,30,60,29,59)(13,37,19,44,39,64,38,61)(14,41,42,56,43,62,25,53)(20,47,26,55,49,57,48,67)(36,63,40,65,52,66,54,68)Polycyclic presentation:
2^9|a^2=i,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325716);