Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| 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 | |
1580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6)(7,18)(8,23,27,40)(9,28)(10,32)(11,16)(12,17)(13,21)(14,26,41,34)(15,29)(19,45,24,35)(20,49)(22,37)(25,39,56,63)(30,42)(31,58)(33,38)(36,51)(43,48,52,47)(44,50)(46,66,54,64)(53,60,57,55)(59,62)(61,67,65,68);(1,2,4,3)(5,7,10,21,12,22,11,33)(6,13,16,18,17,38,32,37)(8,24,34,47,27,19,26,48)(9,20,29,50,31,51,30,59)(14,35,23,52,41,45,40,43)(15,36,28,44,42,49,58,62)(25,54,60,67,56,46,55,68)(39,61,53,66,63,65,57,64);(5,8)(6,14)(7,19)(9,25)(10,26)(11,34)(12,27)(13,35)(15,39)(16,40)(17,41)(18,43)(20,46)(21,47)(22,24)(23,32)(28,57)(29,55)(30,60)(31,56)(33,48)(36,61)(37,52)(38,45)(42,63)(44,64)(49,65)(50,67)(51,54)(53,58)(59,68)(62,66);(1,3,4,2)(5,9,11,30,12,31,10,29)(6,15,32,58,17,42,16,28)(7,20,33,59,22,51,21,50)(8,25,34,60,27,56,26,55)(13,36,37,62,38,49,18,44)(14,39,23,53,41,63,40,57)(19,46,48,68,24,54,47,67)(35,61,52,66,45,65,43,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272242);