Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417859 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1867 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)(7,19)(8,14)(9,29)(10,18)(11,16)(12,17)(13,22)(15,33)(20,46)(21,50)(23,37)(24,38)(25,54)(26,56)(27,45)(28,41)(30,42)(31,44)(32,43)(34,40)(35,53)(36,52)(39,58)(47,51)(48,67)(49,62)(55,61)(57,66)(59,65)(60,64)(63,68);(5,7,10,22,12,23,11,24)(6,13,16,19,17,38,18,37)(8,25,27,53,28,20,34,49)(9,30,31,52,32,51,33,21)(14,35,40,54,41,62,45,46)(15,42,43,50,44,47,29,36)(26,57,58,48,59,68,60,55)(39,63,56,67,64,66,65,61);(1,2,3,4)(5,8)(6,14)(7,20)(9,26)(10,34)(11,27)(12,28)(13,35)(15,39)(16,45)(17,41)(18,40)(19,46)(21,48)(22,53)(23,25)(24,49)(29,56)(30,57)(31,60)(32,59)(33,58)(36,61)(37,54)(38,62)(42,66)(43,65)(44,64)(47,63)(50,67)(51,68)(52,55);(5,9,12,32)(6,15,17,44)(7,21,23,52)(8,26,28,59)(10,33,11,31)(13,36,38,50)(14,39,41,64)(16,29,18,43)(19,47,37,42)(20,48,25,55)(22,51,24,30)(27,60,34,58)(35,61,62,67)(40,65,45,56)(46,63,54,66)(49,57,53,68)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330116);