Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36011 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,29,34)(27,36,38,31)(28,33,41,49)(30,32,43,35)(37,48,40,45)(39,46,52,44)(42,47,54,50)(51,56,53,55);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27)(26,31)(28,40)(29,38)(30,39)(32,44)(33,48)(34,36)(35,46)(37,41)(42,53)(43,52)(45,49)(47,56)(50,55)(51,54);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(30,43)(31,44)(33,47)(34,35)(36,46)(39,52)(42,54)(45,55)(48,56)(49,50)(51,53);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,45)(32,47)(34,49)(35,50)(36,48)(38,40)(39,51)(43,54)(44,55)(46,56)(52,53);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(26,34)(27,38)(28,41)(32,35)(39,52)(42,54)(45,48)(55,56)Polycyclic presentation:
2^9|a^2=i,c^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7110912);