Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13876 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
969 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)(3,14)(4,19,24,38)(5,25)(6,12)(7,13)(8,17)(9,23,35,22)(10,26)(11,28)(15,33,20,29)(16,44)(18,31)(21,51,54,34)(27,36)(30,46)(32,47)(37,55)(39,43,48,42)(40,45)(41,61,50,60)(49,53,59,52)(56,62,58,63)(57,64);(1,3,7,18)(2,8)(4,20)(5,16,27,46)(6,17,28,47)(9,33,35,29)(10,30)(11,14)(12,31)(13,32)(15,24)(19,48,38,39)(21,50)(22,43)(23,42)(25,40)(26,45,55,64)(34,58,51,56)(36,57)(37,44)(41,54)(49,61,59,60)(52,63)(53,62);(1,4)(2,9)(3,15)(5,21)(6,22)(7,24)(8,29)(10,34)(11,19)(12,38)(13,35)(14,39)(16,41)(17,42)(18,20)(23,28)(25,49)(26,52)(27,54)(30,56)(31,48)(32,33)(36,59)(37,51)(40,60)(43,47)(44,58)(45,62)(46,50)(53,55)(57,61)(63,64);(1,5,6,26,7,27,28,55)(2,10,11,25,13,37,12,36)(3,16,17,45,18,46,47,64)(4,21,22,52,24,54,23,53)(8,30,14,40,32,44,31,57)(9,34,19,49,35,51,38,59)(15,41,42,62,20,50,43,63)(29,56,39,60,33,58,48,61)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267692);