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 | #42992 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,37)(28,42)(29,35)(30,36)(31,47)(32,41)(33,43)(34,44)(38,54)(39,49)(40,51)(45,50)(46,53)(48,55)(52,56);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,41)(26,32,36,37)(28,39,44,38)(29,45,31,40)(33,52,53,48)(34,49,42,54)(35,51,47,50)(43,55,46,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,44)(26,33,36,53)(27,38,41,39)(29,43,31,46)(32,48,37,52)(34,47,42,35)(40,55,45,56)(49,51,54,50);(3,8)(5,7)(11,18)(12,20)(14,22)(15,24)(25,28)(26,34)(27,39)(29,46)(30,44)(31,43)(32,49)(33,35)(36,42)(37,54)(38,41)(40,55)(45,56)(47,53)(48,50)(51,52);(2,6)(5,7)(11,18)(14,22)(17,19)(21,23)(25,29,30,31)(26,35,36,47)(27,40,41,45)(28,43,44,46)(32,50,37,51)(33,34,53,42)(38,55,39,56)(48,49,52,54)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,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, 7247157);