Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | #45388 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,36)(28,33)(29,34)(30,32)(31,37)(35,43)(38,48)(39,44)(40,45)(41,50)(42,47)(46,51)(49,54)(52,55)(53,56);(2,6)(3,4)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27)(26,31)(28,40)(29,37)(30,39)(32,46)(33,48)(34,36)(35,44)(38,41)(42,53)(43,51)(45,50)(47,55)(49,56)(52,54);(1,2)(3,8)(4,7)(5,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(27,37)(28,41)(30,43)(31,44)(33,49)(34,35)(36,46)(45,56)(47,50)(48,55)(52,53);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,38)(29,41)(30,42)(31,45)(32,47)(34,50)(35,49)(36,48)(37,40)(39,52)(43,54)(44,55)(46,56)(51,53);(1,4,5,3)(2,7,6,8)(10,18)(15,19)(16,17)(23,24)(26,34)(27,37)(28,41)(32,35)(39,51)(42,54)(45,48)(55,56)Polycyclic presentation:
2^9|a^2=fi,c^2=fi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,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, 7396264);