Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,18,23)(15,25,28,20)(16,21,30,41)(17,32,31,22)(19,24,34,40)(26,43,33,36)(27,37,44,39)(29,42,35,38);(6,10)(11,12)(13,15,18,28)(14,20,23,25)(16,27,30,44)(17,33,31,26)(19,35,34,29)(21,39,41,37)(22,36,32,43)(24,42,40,38);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,18,30)(14,21,23,41)(15,26,28,33)(17,34,31,19)(20,36,25,43)(22,24,32,40)(27,29,44,35)(37,42,39,38);(5,8,9,7)(6,12,10,11)(16,17)(19,34)(21,32)(22,41)(24,40)(26,44)(27,33)(29,35)(30,31)(36,37)(38,42)(39,43);(13,17,18,31)(14,22,23,32)(15,27,28,44)(16,19,30,34)(20,37,25,39)(21,40,41,24)(26,35,33,29)(36,38,43,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491741);