Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418498 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15263 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14)(4,20)(5,26)(6,31)(7,12)(8,13)(9,17)(10,24)(11,30)(15,43)(16,40)(18,35)(19,34)(21,42)(22,51)(23,39)(25,38)(27,33)(28,41)(29,58)(32,45)(36,48)(37,53)(44,63)(46,60)(47,56)(49,55)(50,62)(52,59)(54,57)(61,64);(1,3)(2,9)(4,21)(5,27)(6,18)(7,17)(8,19)(10,36)(11,40)(12,34)(13,35)(14,31)(15,23)(16,28)(20,32)(22,52)(24,45)(25,48)(26,56)(29,47)(30,46)(33,41)(37,59)(38,43)(39,42)(44,55)(49,62)(50,53)(51,61)(54,64)(57,63)(58,60);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,32)(11,37)(12,38)(13,39)(14,42)(16,44)(17,21)(18,48)(19,45)(20,31)(26,57)(27,50)(28,53)(29,54)(30,55)(33,59)(34,36)(35,43)(40,62)(41,51)(46,64)(47,52)(49,58)(56,61)(60,63);(1,5,6,28,8,30,7,29)(2,11,12,41,13,26,31,58)(3,16,17,27,19,47,18,46)(4,22,23,53,25,55,24,54)(9,33,14,40,35,60,34,56)(10,37,38,51,39,57,20,49)(15,44,21,50,45,52,48,64)(32,59,42,62,43,63,36,61)Polycyclic presentation:
2^9|d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325694);