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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,28)(20,25)(21,26)(22,24)(23,29)(27,34)(30,41)(31,35)(32,39)(33,38)(36,42)(37,43)(40,45)(44,46)(47,48);(2,5)(3,7)(10,14)(11,12)(17,19,21,29)(18,23,26,28)(20,30,32,42)(22,31,34,43)(24,37,27,35)(25,36,39,41)(33,44,45,48)(38,47,40,46);(9,11,13,12)(10,15,14,16)(18,24)(19,29)(22,34)(23,35)(25,38)(26,27)(28,37)(30,42)(33,45)(36,46)(39,40)(41,47);(1,3,4,7)(2,6,5,8)(10,14)(15,16)(17,20,21,32)(18,25,26,39)(19,30,29,42)(22,33,34,45)(23,36,28,41)(24,38,27,40)(31,44,43,48)(35,46,37,47);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(19,29)(20,32)(23,28)(25,39)(31,43)(33,45)(35,37)(38,40)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7140996);