Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #23854 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1638 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,33)(21,27)(22,26)(23,28)(25,35)(29,41)(30,44)(31,39)(32,40)(34,38)(36,43)(37,42)(45,47)(46,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,32)(18,24,26,40)(20,34,35,29)(21,31,23,30)(25,41,33,38)(27,39,28,44)(36,45,37,46)(42,48,43,47);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20)(18,25)(19,29)(21,37)(22,35)(23,36)(24,38)(26,33)(27,43)(28,42)(30,46)(31,45)(32,34)(39,48)(40,41)(44,47);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(18,26)(19,30)(20,35)(21,23)(24,39)(29,45)(31,32)(34,46)(38,47)(40,44)(41,48)(42,43);(2,6)(7,8)(17,21,22,23)(18,27,26,28)(19,31,32,30)(20,36,35,37)(24,39,40,44)(25,42,33,43)(29,46,34,45)(38,47,41,48)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7433228);