Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454110 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1453 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)(5,6,12,33,11,17,10,16)(7,13,21,37,22,38,20,36)(8,23,32,14,27,44,26,41)(9,28,30,42,31,15,29,43)(18,45,24,34,48,39,47,60)(19,49,52,61,51,35,50,62)(25,53,56,58,57,64,55,40)(46,65,67,68,66,63,54,59);(1,2,4,3)(5,7,11,22)(6,13,17,38)(8,24,27,47)(9,19,31,51)(10,20,12,21)(14,39,41,45)(15,35,28,49)(16,36,33,37)(18,32,48,26)(23,34,44,60)(25,54,57,67)(29,50,30,52)(40,63,58,65)(42,61,43,62)(46,55,66,56)(53,59,64,68);(2,3)(5,8)(6,14)(7,18)(9,25)(10,32)(11,27)(12,26)(13,34)(15,40)(16,44)(17,41)(19,46)(20,24)(21,47)(22,48)(23,33)(28,58)(29,56)(30,55)(31,57)(35,59)(36,39)(37,45)(38,60)(42,53)(43,64)(49,68)(50,67)(51,66)(52,54)(61,65)(62,63);(1,3,4,2)(5,9)(6,15)(7,19)(8,25)(10,30)(11,31)(12,29)(13,35)(14,40)(16,43)(17,28)(18,46)(20,52)(21,50)(22,51)(23,53)(24,54)(26,56)(27,57)(32,55)(33,42)(34,59)(36,62)(37,61)(38,49)(39,63)(41,58)(44,64)(45,65)(47,67)(48,66)(60,68)Polycyclic presentation:
2^9|a^2=fi,b^2=ei,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362272);