Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1843 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)(7,19)(8,25)(9,31)(10,17)(11,16)(12,18)(13,22)(14,28)(15,34)(20,49)(21,53)(23,39)(24,38)(26,47)(27,58)(29,43)(30,42)(32,44)(33,46)(35,45)(36,52)(37,55)(40,51)(41,61)(48,54)(50,64)(56,60)(57,65)(59,66)(62,63)(67,68);(1,2,3,4)(5,7,10,22,12,24,11,23)(6,13,16,19,18,39,17,38)(8,26,28,52,30,51,29,20)(9,32,33,55,35,54,34,21)(14,40,25,36,43,47,42,49)(15,44,45,53,46,48,31,37)(27,50,59,62,61,68,60,57)(41,65,66,67,58,63,56,64);(5,8,10,28,12,30,11,29)(6,14,16,25,18,43,17,42)(7,20,22,26,24,52,23,51)(9,27,34,60,35,61,33,59)(13,36,19,47,39,49,38,40)(15,41,31,56,46,58,45,66)(21,50,54,57,55,68,32,62)(37,63,48,67,53,65,44,64);(2,4)(5,9)(6,15)(7,21)(8,27)(10,34)(11,33)(12,35)(13,37)(14,41)(16,31)(17,45)(18,46)(19,48)(20,50)(22,54)(23,32)(24,55)(25,56)(26,57)(28,60)(29,59)(30,61)(36,63)(38,44)(39,53)(40,64)(42,66)(43,58)(47,67)(49,65)(51,62)(52,68)Polycyclic presentation:
2^9|b^2=e,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330119);