Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500685 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24208 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1150 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)(3,16)(4,22)(5,14)(6,30)(7,31)(8,13)(9,15)(10,19)(11,24)(12,28)(17,47)(18,46)(20,53)(21,37)(23,56)(25,44)(26,55)(27,40)(29,45)(32,41)(33,38)(34,60)(35,52)(36,51)(39,48)(42,57)(43,58)(49,61)(50,64)(54,63)(59,62);(1,3,6,19,9,21,8,33)(2,10,13,16,15,38,30,37)(4,23,25,17,27,49,26,50)(5,18,28,51,29,52,7,20)(11,39,41,63,43,62,42,34)(12,35,14,36,31,46,45,53)(22,47,55,56,40,64,44,61)(24,54,57,48,58,60,32,59);(1,4)(2,11)(3,17)(5,24)(6,25)(7,32)(8,26)(9,27)(10,34)(12,40)(13,41)(14,44)(15,43)(16,39)(18,48)(19,49)(20,54)(21,50)(22,31)(23,33)(28,57)(29,58)(30,42)(35,56)(36,64)(37,62)(38,63)(45,55)(46,61)(47,53)(51,60)(52,59);(1,5,6,28,9,29,8,7)(2,12,13,14,15,31,30,45)(3,18,19,51,21,52,33,20)(4,24,25,57,27,58,26,32)(10,35,16,36,38,46,37,53)(11,40,41,44,43,22,42,55)(17,48,49,60,50,59,23,54)(34,56,39,64,63,61,62,47)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,g^b=gi,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407881);