Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417859 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16)(7,17,22,44)(8,13,25,37)(9,14,28,40)(10,29,18,15)(12,31,34,21)(19,42,48,65)(20,43,49,38)(23,51,54,35)(24,55,45,36)(26,57,50,33)(27,41,30,39)(32,58,60,47)(46,61,66,64)(52,59,67,68)(53,62,56,63);(1,2)(3,4)(5,7,18,31,11,22,10,21)(6,12,29,44,16,34,15,17)(8,19,45,58,25,48,24,47)(9,26,30,20,28,50,27,49)(13,32,55,65,37,60,36,42)(14,38,41,33,40,43,39,57)(23,52,56,46,54,67,53,66)(35,61,63,59,51,64,62,68);(2,4)(5,8)(6,13)(7,19)(9,23)(10,24)(11,25)(12,32)(14,35)(15,36)(16,37)(17,42)(18,45)(20,46)(21,47)(22,48)(26,52)(27,53)(28,54)(29,55)(30,56)(31,58)(33,59)(34,60)(38,61)(39,62)(40,51)(41,63)(43,64)(44,65)(49,66)(50,67)(57,68);(5,9)(6,14)(7,20)(8,23)(10,30)(11,28)(12,33)(13,35)(15,41)(16,40)(17,43)(18,27)(19,46)(21,50)(22,49)(24,56)(25,54)(26,31)(29,39)(32,59)(34,57)(36,63)(37,51)(38,44)(42,64)(45,53)(47,67)(48,66)(52,58)(55,62)(60,68)(61,65)Polycyclic presentation:
2^9|a^2=i,b^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325058);