Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42882 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(6,10)(7,8)(13,14,19,25)(15,26,30,21)(16,22,33,42)(17,34,20,23)(18,24,31,44)(27,43,35,37)(28,38,36,40)(29,41,32,39);(5,6)(7,12)(8,11)(9,10)(13,15,19,30)(14,21,25,26)(16,31,33,18)(17,28,20,36)(22,24,42,44)(23,38,34,40)(27,32,35,29)(37,39,43,41);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,32,19,33,20,29)(14,22,34,39,25,42,23,41)(15,27,36,18,30,35,28,31)(21,37,38,44,26,43,40,24);(5,8,9,7)(6,11,10,12)(13,17,19,20)(14,23,25,34)(15,28,30,36)(16,32,33,29)(18,27,31,35)(21,38,26,40)(22,41,42,39)(24,37,44,43);(13,18,20,27,19,31,17,35)(14,24,23,43,25,44,34,37)(15,29,28,33,30,32,36,16)(21,39,40,22,26,41,38,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6718018);