Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396363 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14732 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1380 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)(3,14)(4,10)(5,11,24,39)(6,26)(7,12)(8,13)(9,17)(15,33)(16,40,45,56)(18,31)(19,46)(20,48,50,34)(21,36)(22,37)(23,51,25,38)(27,32)(28,35)(29,42)(30,52,55,44)(41,61,63,58)(43,57)(47,54,62,64)(49,59,53,60);(1,3,6,17,8,18,7,27)(2,9,12,14,13,32,26,31)(4,19,21,15,22,42,28,43)(5,16,23,44,24,45,25,52)(10,33,35,46,37,57,36,29)(11,30,38,40,39,55,51,56)(20,47,49,41,50,62,53,63)(34,58,59,64,48,61,60,54);(1,4,8,22)(2,10,13,37)(3,15,18,43)(5,20,24,50)(6,21,7,28)(9,29,32,46)(11,34,39,48)(12,35,26,36)(14,33,31,57)(16,41,45,63)(17,42,27,19)(23,49,25,53)(30,54,55,64)(38,59,51,60)(40,58,56,61)(44,62,52,47);(1,5,6,23,8,24,7,25)(2,11,12,38,13,39,26,51)(3,16,17,44,18,45,27,52)(4,20,21,49,22,50,28,53)(9,30,14,40,32,55,31,56)(10,34,35,59,37,48,36,60)(15,41,42,62,43,63,19,47)(29,54,33,58,46,64,57,61)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303560);