Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C24 |
| commutator, socle | C22 | C23×C16 |
| prime radical | #532 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,9,16,10,17,28,43,11,18,29,44,31,45,56,65,12,15,30,40,32,41,57,63,33,42,34,59,58,64,60,68)(7,13,20,35,21,36,49,61,22,8,23,24,50,25,51,52,19,26,46,53,47,54,66,67,48,27,14,37,38,55,39,62);(2,4)(5,7)(6,13)(8,18)(9,20)(10,21)(11,22)(12,19)(14,34)(15,26)(16,35)(17,36)(23,29)(24,44)(25,45)(27,42)(28,49)(30,46)(31,50)(32,47)(33,48)(37,59)(38,58)(39,60)(40,53)(41,54)(43,61)(51,56)(52,65)(55,64)(57,66)(62,68)(63,67);(5,8,9,24,10,25,28,52,11,26,29,53,31,54,56,67,12,27,30,37,32,55,57,62,33,13,34,35,58,36,60,61)(6,14,16,38,17,39,43,7,18,20,44,21,45,49,65,22,15,23,40,50,41,51,63,19,42,46,59,47,64,66,68,48);(6,15)(7,19)(14,23)(16,40)(17,41)(18,42)(20,46)(21,47)(22,48)(38,50)(39,51)(43,63)(44,59)(45,64)(49,66)(65,68)Polycyclic presentation:
2^9|a^2=f,c^2=f,f^2=g,g^2=h,h^2=i,b^a=be,c^a=ci,d^a=di,d^b=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72231859, #72320813, #72320831, … |
| maximal normal subgroup of | #72231859, #72320813, #72320831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419941);