Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455391 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,15,18,33,20,9,19,34)(4,21,24,10,25,41,30,37)(5,26,27,39,29,11,28,40)(16,44,22,42,46,35,50,31)(17,32,47,59,49,43,48,58)(23,51,53,36,54,62,55,38)(45,61,52,57,63,60,64,56);(1,3,8,20)(2,9,12,15)(4,22,25,50)(5,17,29,49)(6,19,7,18)(10,35,37,44)(11,32,26,43)(13,34,14,33)(16,30,46,24)(21,31,41,42)(23,52,54,64)(27,48,28,47)(36,60,38,61)(39,59,40,58)(45,55,63,53)(51,56,62,57);(1,4)(2,10,12,37)(3,16)(5,23)(6,24)(7,30)(8,25)(9,31,15,42)(11,38,26,36)(13,21,14,41)(17,45)(18,22)(19,50)(20,46)(27,53)(28,55)(29,54)(32,57,43,56)(33,35,34,44)(39,62,40,51)(47,52)(48,64)(49,63)(58,61,59,60);(1,5)(2,11,12,26)(3,17)(4,23)(6,27)(7,28)(8,29)(9,32,15,43)(10,36,37,38)(13,39,14,40)(16,45)(18,47)(19,48)(20,49)(21,51,41,62)(22,52)(24,53)(25,54)(30,55)(31,56,42,57)(33,58,34,59)(35,60,44,61)(46,63)(50,64)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362588);