Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,28,31,40)(20,32,34,26)(21,27,23,36)(25,37,38,30)(29,44,46,43)(33,42,35,41)(39,47,48,45);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,24)(25,38)(26,32)(27,36)(28,40)(39,48)(41,42)(43,44);(11,15)(13,16)(17,19,22,31)(18,25,24,38)(20,29,34,46)(21,30,23,37)(26,39,32,48)(27,40,36,28)(33,45,35,47)(41,43,42,44);(3,8)(5,7)(10,14)(11,15)(17,20)(18,26)(19,29)(21,35)(22,34)(23,33)(24,32)(25,39)(27,42)(28,43)(30,47)(31,46)(36,41)(37,45)(38,48)(40,44);(2,6)(5,7)(11,15)(13,16)(17,21,22,23)(18,27,24,36)(19,30,31,37)(20,33,34,35)(25,40,38,28)(26,41,32,42)(29,45,46,47)(39,43,48,44)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7107593);