Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397514 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1970 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)(7,19)(8,25)(9,30)(10,18)(11,16)(12,17)(13,22)(14,28)(15,32)(20,35)(21,36)(23,37)(24,38)(26,54)(27,56)(29,43)(31,42)(33,41)(34,40)(39,58)(44,47)(45,50)(46,65)(48,59)(49,62)(51,52)(53,57)(55,64)(60,63)(61,66)(67,68);(1,2,3,4)(5,7,10,22,12,23,11,24)(6,13,16,19,17,38,18,37)(8,26,28,51,29,20,34,47)(9,21,31,48,33,49,32,50)(14,35,25,52,40,54,43,44)(15,36,41,45,42,62,30,59)(27,55,57,67,58,46,60,66)(39,61,53,65,56,68,63,64);(5,8)(6,14)(7,20)(9,27)(10,34)(11,28)(12,29)(13,35)(15,39)(16,43)(17,40)(18,25)(19,44)(21,46)(22,51)(23,26)(24,47)(30,53)(31,60)(32,57)(33,58)(36,61)(37,52)(38,54)(41,63)(42,56)(45,64)(48,67)(49,55)(50,66)(59,65)(62,68);(5,9,10,31,12,33,11,32)(6,15,16,41,17,42,18,30)(7,21,22,48,23,49,24,50)(8,27,34,60,29,58,28,57)(13,36,19,45,38,62,37,59)(14,39,43,63,40,56,25,53)(20,46,51,67,26,55,47,66)(35,61,44,64,54,68,52,65)Polycyclic presentation:
2^9|b^2=e,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319051);