Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2761 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16402 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
741 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)(3,13)(4,18)(5,22,24,10)(6,25)(7,12)(8,16)(9,20)(11,26)(14,36)(15,28,41,34)(17,29)(19,47,49,45)(21,46)(23,32,53,52)(27,54)(30,48,51,63)(31,50)(33,39)(35,43)(37,60,40,59)(38,44)(42,58,61,57)(55,64,56,62);(1,3,6,17,7,16,26,43)(2,8,11,29,12,13,25,35)(4,14,20,39,21,38,50,54)(5,15,23,42,24,41,53,61)(9,27,18,44,31,33,46,36)(10,28,32,57,22,34,52,58)(19,37,48,62,49,40,63,64)(30,55,45,60,51,56,47,59);(1,4,6,20,7,21,26,50)(2,9,11,18,12,31,25,46)(3,14,17,39,16,38,43,54)(5,19,23,48,24,49,53,63)(8,27,29,44,13,33,35,36)(10,30,32,45,22,51,52,47)(15,40,42,64,41,37,61,62)(28,56,57,59,34,55,58,60);(1,5)(2,10)(3,15)(4,19)(6,23)(7,24)(8,28)(9,30)(11,32)(12,22)(13,34)(14,37)(16,41)(17,42)(18,45)(20,48)(21,49)(25,52)(26,53)(27,55)(29,57)(31,51)(33,56)(35,58)(36,59)(38,40)(39,62)(43,61)(44,60)(46,47)(50,63)(54,64)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368989);