Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23897 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,25)(3,9,17,30,18,31,16,45)(4,19,22,10,21,35,26,34)(5,11,23,38,24,37,27,36)(14,39,42,28,41,48,46,32)(15,29,43,57,44,56,47,55)(20,50,53,59,51,49,52,33)(40,61,64,58,62,60,63,54);(1,3,7,17,8,18,6,16)(2,9,12,30,13,31,25,45)(4,14,22,42,21,41,26,46)(5,15,23,43,24,44,27,47)(10,32,35,39,34,28,19,48)(11,29,38,57,37,56,36,55)(20,40,53,64,51,62,52,63)(33,58,50,60,59,54,49,61);(1,4)(2,10)(3,14)(5,20)(6,26)(7,22)(8,21)(9,28)(11,33)(12,35)(13,34)(15,40)(16,46)(17,42)(18,41)(19,25)(23,53)(24,51)(27,52)(29,54)(30,48)(31,32)(36,49)(37,59)(38,50)(39,45)(43,64)(44,62)(47,63)(55,60)(56,58)(57,61);(1,5)(2,11)(3,15)(4,20)(6,27)(7,23)(8,24)(9,29)(10,33)(12,38)(13,37)(14,40)(16,47)(17,43)(18,44)(19,49)(21,51)(22,53)(25,36)(26,52)(28,54)(30,57)(31,56)(32,58)(34,59)(35,50)(39,60)(41,62)(42,64)(45,55)(46,63)(48,61)Polycyclic presentation:
2^9|a^2=h,b^2=h,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^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285536, #72285720, #72285753, … |
| maximal normal subgroup of | #72285536, #72285720, #72285753, … |
| maximal quotient of | C2×#364537 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271733);