Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13175 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1943 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1943 |
| prime radical | C24 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
595 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)(3,15)(4,21,25,53)(5,11)(6,28)(7,12)(8,14)(9,18)(10,24,38,58)(13,30)(16,44,47,50)(17,34)(19,41)(20,39)(22,54,45,64)(23,51,26,52)(27,42)(29,33)(31,40)(32,55,62,57)(35,46,63,61)(36,56,37,59)(43,49)(48,60);(1,3)(2,9)(4,22,25,45)(5,17)(6,29)(7,18)(8,20)(10,35,38,63)(11,15)(12,39)(13,41)(14,34)(16,26,47,23)(19,31)(21,50,53,44)(24,57,58,55)(27,49)(28,60)(30,48)(32,36,62,37)(33,42)(40,43)(46,56,61,59)(51,54,52,64);(1,4,6,24,7,26,30,59,8,25,31,58,5,23,27,56)(2,10,13,21,11,36,28,51,14,38,42,53,12,37,40,52)(3,16,19,46,18,22,49,55,20,47,29,61,17,45,48,57)(9,32,33,54,15,35,43,44,34,62,41,64,39,63,60,50);(1,5,8,7)(2,11,14,12)(3,17,20,18)(4,23,25,26)(6,27,31,30)(9,15,34,39)(10,36,38,37)(13,28,42,40)(16,45,47,22)(19,48,29,49)(21,51,53,52)(24,56,58,59)(32,35,62,63)(33,43,41,60)(44,64,50,54)(46,57,61,55)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=i,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285749, #72285848, #72286027, … |
| maximal normal subgroup of | #72285749, #72285848, #72286027, … |
| maximal quotient of | #72310163, #72322817, #72342624 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402243);