Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2521 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)(7,13)(8,24)(9,29)(10,17)(11,16)(12,18)(14,27)(15,30)(19,39)(20,48)(21,37)(22,36)(23,38)(25,52)(26,40)(28,41)(31,43)(32,42)(33,44)(34,46)(35,49)(45,59)(47,61)(50,60)(51,62)(53,56)(54,63)(55,58)(57,64)(65,67)(66,68);(1,2,3,4)(5,7,12,23)(6,13,18,38)(8,25,28,46)(9,20,31,51)(10,21,11,22)(14,39,42,61)(15,35,44,60)(16,36,17,37)(19,32,47,27)(24,52,41,34)(26,54,57,65)(29,48,43,62)(30,49,33,50)(40,63,64,67)(45,56,66,55)(53,68,58,59);(5,8)(6,14)(7,19)(9,26)(10,32)(11,27)(12,28)(13,34)(15,40)(16,41)(17,24)(18,42)(20,45)(21,25)(22,46)(23,47)(29,58)(30,56)(31,57)(33,55)(35,59)(36,61)(37,39)(38,52)(43,53)(44,64)(48,67)(49,54)(50,65)(51,66)(60,68)(62,63);(5,9,12,31)(6,15,18,44)(7,20,23,51)(8,26,28,57)(10,30,11,33)(13,35,38,60)(14,40,42,64)(16,29,17,43)(19,45,47,66)(21,49,22,50)(24,53,41,58)(25,54,46,65)(27,55,32,56)(34,59,52,68)(36,48,37,62)(39,63,61,67)Polycyclic presentation:
2^9|b^2=ei,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301544);