Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454977 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,10,16,11,15,32,42)(7,17,20,43,21,12,30,35)(8,22,25,44,26,13,31,38)(9,27,28,56,29,14,39,41)(18,33,23,36,48,45,51,52)(19,34,49,61,50,46,57,66)(24,40,54,65,55,37,58,63)(47,59,53,62,67,60,68,64);(5,7,11,21)(6,12,15,17)(8,23,26,51)(9,19,29,50)(10,30,32,20)(13,36,22,52)(14,34,27,46)(16,43,42,35)(18,31,48,25)(24,53,55,68)(28,57,39,49)(33,44,45,38)(37,62,40,64)(41,66,56,61)(47,58,67,54)(59,65,60,63);(2,4)(5,8)(6,13)(7,18)(9,24)(10,31)(11,26)(12,33)(14,40)(15,22)(16,44)(17,45)(19,47)(20,51)(21,48)(23,30)(25,32)(27,37)(28,58)(29,55)(34,60)(35,52)(36,43)(38,42)(39,54)(41,63)(46,59)(49,68)(50,67)(53,57)(56,65)(61,62)(64,66);(5,9,11,29)(6,14)(7,19,21,50)(8,24,26,55)(10,28,32,39)(12,34)(13,37)(15,27)(16,41)(17,46)(18,47,48,67)(20,49,30,57)(22,40)(23,53,51,68)(25,54,31,58)(33,59)(35,61)(36,62)(38,63)(42,56)(43,66)(44,65)(45,60)(52,64)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367502);