Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394358 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
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 | |
1835 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)(5,6,11,15)(7,12,19,34)(8,13,23,36)(9,25,27,39)(10,28,30,16)(14,29,38,26)(17,31,21,48)(18,42,43,60)(20,46,45,33)(22,51,52,62)(24,55,54,37)(32,47,59,44)(35,56,61,53)(40,63,49,68)(41,65,50,58)(57,66,67,64);(5,7)(6,12)(8,21,23,17)(9,18)(10,20)(11,19)(13,31,36,48)(14,32)(15,34)(16,33)(22,49,52,40)(24,50,54,41)(25,42)(26,44)(27,43)(28,46)(29,47)(30,45)(35,57,61,67)(37,58,55,65)(38,59)(39,60)(51,63,62,68)(53,66,56,64);(1,2,3,4)(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,31)(14,35)(15,36)(16,37)(18,40)(19,21)(20,41)(25,51)(26,53)(27,52)(28,55)(29,56)(30,54)(32,57)(33,58)(34,48)(38,61)(39,62)(42,63)(43,49)(44,64)(45,50)(46,65)(47,66)(59,67)(60,68);(5,9,11,27)(6,14,15,38)(7,18,19,43)(8,22,23,52)(10,29,30,26)(12,32,34,59)(13,35,36,61)(16,39,28,25)(17,40,21,49)(20,47,45,44)(24,56,54,53)(31,57,48,67)(33,60,46,42)(37,62,55,51)(41,66,50,64)(58,68,65,63)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305113);