Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1224 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)(8,24,27,37)(9,28)(10,16)(11,17)(12,21)(13,26,38,44)(14,30)(15,29)(19,35,48,32)(20,49)(22,34)(23,41)(25,36,55,61)(31,39)(33,52)(40,57)(42,47,60,46)(43,50)(45,65,68,63)(51,62)(53,64,56,54)(58,66,59,67);(5,7,10,21,11,23,29,22)(6,12,15,18,17,34,16,41)(8,19,44,47,27,48,26,46)(9,20,30,50,31,52,57,51)(13,35,24,42,38,32,37,60)(14,33,28,43,40,49,39,62)(25,45,64,67,55,68,54,66)(36,59,53,63,61,58,56,65);(1,2,3,4)(5,8,11,27)(6,13)(7,19)(9,25,31,55)(10,26,29,44)(12,32,34,35)(14,36)(15,37)(16,24)(17,38)(18,42,41,60)(20,45)(21,46)(22,47)(23,48)(28,56)(30,54,57,64)(33,58,49,59)(39,53)(40,61)(43,63,62,65)(50,66)(51,67)(52,68);(5,9,29,57,11,31,10,30)(6,14,16,39,17,40,15,28)(7,20,22,51,23,52,21,50)(8,25,44,64,27,55,26,54)(12,33,41,62,34,49,18,43)(13,36,24,53,38,61,37,56)(19,45,47,67,48,68,46,66)(32,58,60,65,35,59,42,63)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=ef,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,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272973);