Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1532 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)(5,6,10,16,12,18,11,17)(7,13,21,36,22,37,33,44)(8,23,26,41,28,40,27,14)(9,29,30,43,31,15,32,42)(19,38,24,52,46,62,51,34)(20,47,48,61,50,35,49,60)(25,53,55,39,56,64,57,58)(45,63,65,59,66,68,54,67);(1,2)(3,4)(5,7)(6,13)(8,24)(9,20)(10,21)(11,33)(12,22)(14,38)(15,35)(16,36)(17,44)(18,37)(19,27)(23,52)(25,54)(26,46)(28,51)(29,47)(30,48)(31,50)(32,49)(34,40)(39,63)(41,62)(42,60)(43,61)(45,55)(53,67)(56,65)(57,66)(58,68)(59,64);(5,8,12,28)(6,14,18,41)(7,19,22,46)(9,25,31,56)(10,27,11,26)(13,34,37,52)(15,39,29,58)(16,40,17,23)(20,45,50,66)(21,51,33,24)(30,57,32,55)(35,59,47,67)(36,62,44,38)(42,53,43,64)(48,54,49,65)(60,63,61,68);(1,3,4,2)(5,9,12,31)(6,15,18,29)(7,20,22,50)(8,25,28,56)(10,32,11,30)(13,35,37,47)(14,39,41,58)(16,43,17,42)(19,45,46,66)(21,49,33,48)(23,53,40,64)(24,54,51,65)(26,57,27,55)(34,59,52,67)(36,61,44,60)(38,63,62,68)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326976);