Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,11,18,12,17,10,16)(7,19,24,13,23,36,22,37)(8,25,28,39,29,14,27,40)(9,30,32,15,33,42,31,41)(20,43,48,52,47,60,46,34)(21,35,51,44,50,62,49,61)(26,54,56,38,57,58,55,53)(45,59,66,68,65,67,64,63);(1,2,3,4)(5,7)(6,13)(8,20)(9,21)(10,24)(11,22)(12,23)(14,34)(15,35)(16,36)(17,37)(18,19)(25,52)(26,45)(27,48)(28,46)(29,47)(30,44)(31,51)(32,49)(33,50)(38,59)(39,43)(40,60)(41,62)(42,61)(53,67)(54,68)(55,66)(56,64)(57,65)(58,63);(5,8)(6,14)(7,20)(9,26)(10,28)(11,27)(12,29)(13,34)(15,38)(16,40)(17,25)(18,39)(19,43)(21,45)(22,48)(23,47)(24,46)(30,58)(31,56)(32,55)(33,57)(35,59)(36,60)(37,52)(41,53)(42,54)(44,63)(49,66)(50,65)(51,64)(61,68)(62,67);(5,9,12,33)(6,15,17,41)(7,21,23,50)(8,26,29,57)(10,32,11,31)(13,35,37,62)(14,38,25,53)(16,42,18,30)(19,44,36,61)(20,45,47,65)(22,51,24,49)(27,56,28,55)(34,59,52,67)(39,58,40,54)(43,63,60,68)(46,66,48,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299920);