Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45316 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(10,13)(11,15)(17,18)(19,24)(20,30)(21,26)(22,37)(23,27)(25,32)(28,41)(29,38)(31,47)(33,40)(34,48)(35,43)(36,44)(39,46)(42,45);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,29,36,35)(25,30,41,40)(26,38,44,43)(31,45,34,46)(39,47,42,48);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,32,23,33,19,28)(18,25,24,30,27,41,37,40)(21,34,35,45,36,31,29,46)(26,42,38,48,44,39,43,47);(3,8)(6,7)(10,13)(14,16)(17,21,23,36)(18,26,27,44)(19,29,22,35)(20,31,33,34)(24,38,37,43)(25,39,41,42)(28,45,32,46)(30,47,40,48);(9,11,12,15)(10,14,13,16)(17,19,23,22)(18,24,27,37)(20,32,33,28)(21,35,36,29)(25,40,41,30)(26,43,44,38)(31,45,34,46)(39,47,42,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgh,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7102596);