Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14825 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,29)(3,14,18,34,19,33,17,9)(4,20,24,10,25,36,23,37)(5,26,30,58,28,11,27,38)(15,41,48,49,21,31,43,39)(16,44,46,32,47,40,45,56)(22,52,54,57,55,50,53,35)(42,59,64,60,51,61,62,63);(1,3,6,17,7,19,8,18)(2,9,12,14,13,34,29,33)(4,21,23,48,25,15,24,43)(5,16,27,45,28,47,30,46)(10,31,36,39,37,41,20,49)(11,32,38,40,26,56,58,44)(22,51,53,64,55,42,54,62)(35,59,52,60,57,61,50,63);(1,4)(2,10)(3,15)(5,22)(6,23)(7,25)(8,24)(9,31)(11,35)(12,36)(13,37)(14,39)(16,42)(17,43)(18,48)(19,21)(20,29)(26,57)(27,53)(28,55)(30,54)(32,59)(33,49)(34,41)(38,52)(40,60)(44,63)(45,62)(46,64)(47,51)(50,58)(56,61);(1,5,6,27,7,28,8,30)(2,11,12,38,13,26,29,58)(3,16,17,45,19,47,18,46)(4,22,23,53,25,55,24,54)(9,32,14,40,34,56,33,44)(10,35,36,52,37,57,20,50)(15,42,43,62,21,51,48,64)(31,59,39,60,41,61,49,63)Polycyclic presentation:
2^9|a^2=eg,b^2=e,d^2=e,e^2=i,f^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,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285536, #72285784, #72285960, … |
| maximal normal subgroup of | #72285536, #72285784, #72285960, … |
| maximal quotient of | C2×#401939 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309135);