Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622976 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,4)(5,6,11,16)(7,17,20,40)(8,13,23,34)(9,26,28,54)(10,29,27,15)(12,30,32,21)(14,37,38,18)(19,39,45,61)(22,49,48,42)(24,52,50,67)(25,53,51,36)(31,55,57,46)(33,43,41,56)(35,60,59,44)(47,66,65,63)(58,64,62,68);(5,7,18,43,10,22,28,30,11,20,37,56,27,48,9,21)(6,12,26,42,15,33,38,40,16,32,54,49,29,41,14,17)(8,19,44,64,25,47,50,55,23,45,60,68,51,65,24,46)(13,31,52,63,36,58,59,61,34,57,67,66,53,62,35,39);(1,2,3,4)(5,8)(6,13)(7,19)(9,24)(10,25)(11,23)(12,31)(14,35)(15,36)(16,34)(17,39)(18,44)(20,45)(21,46)(22,47)(26,52)(27,51)(28,50)(29,53)(30,55)(32,57)(33,58)(37,60)(38,59)(40,61)(41,62)(42,63)(43,64)(48,65)(49,66)(54,67)(56,68);(8,23)(13,34)(19,45)(24,50)(25,51)(31,57)(35,59)(36,53)(39,61)(44,60)(46,55)(47,65)(52,67)(58,62)(63,66)(64,68);(7,20)(12,32)(17,40)(19,45)(21,30)(22,48)(31,57)(33,41)(39,61)(42,49)(43,56)(46,55)(47,65)(58,62)(63,66)(64,68)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=g,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72343356, #72343358, #72343374, … |
| maximal normal subgroup of | #72343356, #72343358, #72343374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530396);