Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14015 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1557 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,26)(3,9,17,30,18,31,16,45)(4,19,23,34,21,49,22,10)(5,11,24,37,25,36,27,35)(14,38,42,32,40,47,41,28)(15,29,43,57,44,56,46,55)(20,50,52,33,51,48,53,59)(39,61,63,54,62,60,64,58);(1,3,7,17,8,18,6,16)(2,9,12,30,13,31,26,45)(4,14,23,42,21,40,22,41)(5,15,24,43,25,44,27,46)(10,32,19,47,34,28,49,38)(11,29,37,57,36,56,35,55)(20,39,52,63,51,62,53,64)(33,58,48,61,59,54,50,60);(1,4)(2,10,13,34)(3,14)(5,20)(6,22)(7,23)(8,21)(9,28,31,32)(11,33,36,59)(12,19,26,49)(15,39)(16,41)(17,42)(18,40)(24,52)(25,51)(27,53)(29,54,56,58)(30,38,45,47)(35,50,37,48)(43,63)(44,62)(46,64)(55,61,57,60);(1,5)(2,11)(3,15)(4,20)(6,27)(7,24)(8,25)(9,29)(10,33)(12,37)(13,36)(14,39)(16,46)(17,43)(18,44)(19,48)(21,51)(22,53)(23,52)(26,35)(28,54)(30,57)(31,56)(32,58)(34,59)(38,60)(40,62)(41,64)(42,63)(45,55)(47,61)(49,50)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^c=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285582, #72285720, #72285784, … |
| maximal normal subgroup of | #72285582, #72285720, #72285784, … |
| maximal quotient of | C2×#368088 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275284);