Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443050 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(5,6,12,17,11,18,10,16)(7,19,23,13,24,42,22,35)(8,25,31,38,28,37,27,14)(9,15,32,41,29,39,30,40)(20,45,48,43,49,33,47,61)(21,44,52,34,50,62,51,58)(26,53,56,59,54,60,55,36)(46,64,67,63,65,57,66,68);(2,3)(5,7)(6,13)(8,20)(9,21)(10,23)(11,24)(12,22)(14,33)(15,34)(16,42)(17,19)(18,35)(25,43)(26,46)(27,48)(28,49)(29,50)(30,52)(31,47)(32,51)(36,57)(37,61)(38,45)(39,58)(40,62)(41,44)(53,63)(54,65)(55,67)(56,66)(59,64)(60,68);(1,2,4,3)(5,8,12,31,11,28,10,27)(6,14,16,37,18,38,17,25)(7,20,22,47,24,49,23,48)(9,26,32,56,29,54,30,55)(13,33,42,61,35,45,19,43)(15,36,40,60,39,59,41,53)(21,46,51,66,50,65,52,67)(34,57,62,68,58,64,44,63);(1,3,4,2)(5,9)(6,15)(7,21)(8,26)(10,30)(11,29)(12,32)(13,34)(14,36)(16,40)(17,41)(18,39)(19,44)(20,46)(22,51)(23,52)(24,50)(25,53)(27,55)(28,54)(31,56)(33,57)(35,58)(37,60)(38,59)(42,62)(43,63)(45,64)(47,66)(48,67)(49,65)(61,68);(9,29)(15,39)(21,50)(26,54)(30,32)(34,58)(36,59)(40,41)(44,62)(46,65)(51,52)(53,60)(55,56)(57,64)(63,68)(66,67)Polycyclic presentation:
2^9|a^2=fi,c^2=ghi,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372794);