Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| 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 | |
1746 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)(7,18)(8,23)(9,28,31,40)(10,32)(11,16)(12,17)(13,21)(14,25)(15,30,41,29)(19,44)(20,48,51,35)(22,36)(24,38,56,62)(26,39)(27,53)(33,37)(34,47)(42,46)(43,49,58,50)(45,65,68,63)(52,55,59,54)(57,61)(60,67,64,66);(1,2,3,4)(5,7,10,21,12,22,11,33)(6,13,16,18,17,37,32,36)(8,19,25,46,27,47,26,57)(9,20,30,50,31,51,29,49)(14,34,23,42,39,44,53,61)(15,35,40,43,41,48,28,58)(24,45,55,67,56,68,54,66)(38,60,52,63,62,64,59,65);(5,8,11,26,12,27,10,25)(6,14,32,53,17,39,16,23)(7,19,33,57,22,47,21,46)(9,24,29,54,31,56,30,55)(13,34,36,61,37,44,18,42)(15,38,28,59,41,62,40,52)(20,45,49,66,51,68,50,67)(35,60,58,65,48,64,43,63);(5,9)(6,15)(7,20)(8,24)(10,30)(11,29)(12,31)(13,35)(14,38)(16,40)(17,41)(18,43)(19,45)(21,50)(22,51)(23,52)(25,55)(26,54)(27,56)(28,32)(33,49)(34,60)(36,58)(37,48)(39,62)(42,63)(44,64)(46,67)(47,68)(53,59)(57,66)(61,65)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^b=gi,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, 300287);