Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1124 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)(5,6,10,16,11,15,32,42)(7,17,20,43,21,12,30,35)(8,22,25,44,26,13,31,38)(9,27,28,56,29,14,39,41)(18,33,23,36,47,45,50,51)(19,40,48,65,49,34,57,60)(24,37,54,63,55,52,58,66)(46,59,53,61,67,64,68,62);(1,2,3,4)(5,7)(6,12)(8,23)(9,19)(10,30)(11,21)(13,36)(14,40)(15,17)(16,43)(18,25)(20,32)(22,51)(24,53)(26,50)(27,34)(28,57)(29,49)(31,47)(33,38)(35,42)(37,62)(39,48)(41,60)(44,45)(46,54)(52,61)(55,68)(56,65)(58,67)(59,66)(63,64);(5,8,11,26)(6,13,15,22)(7,18,21,47)(9,24,29,55)(10,31,32,25)(12,33,17,45)(14,37,27,52)(16,44,42,38)(19,46,49,67)(20,50,30,23)(28,58,39,54)(34,59,40,64)(35,51,43,36)(41,66,56,63)(48,68,57,53)(60,62,65,61);(2,4)(5,9,11,29)(6,14)(7,19,21,49)(8,24,26,55)(10,28,32,39)(12,34)(13,37)(15,27)(16,41)(17,40)(18,46,47,67)(20,48,30,57)(22,52)(23,53,50,68)(25,54,31,58)(33,59)(35,60)(36,61)(38,63)(42,56)(43,65)(44,66)(45,64)(51,62)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=i,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,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, 343666);