Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,27)(16,32)(17,23)(18,24)(19,25)(20,26)(21,29)(22,28)(30,39)(31,38)(33,40)(34,42)(35,41)(36,43)(37,44);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,30,20,31,18,29)(14,21,24,38,26,39,25,27)(16,33,28,37,35,17,34,36)(22,40,32,43,42,23,41,44);(6,10)(7,11)(13,16,18,34,20,35,19,28)(14,22,25,41,26,42,24,32)(15,17,29,37,31,33,30,36)(21,23,27,43,39,40,38,44);(7,11)(9,12)(13,17)(14,23)(15,28)(16,29)(18,37)(19,36)(20,33)(21,32)(22,27)(24,44)(25,43)(26,40)(30,35)(31,34)(38,42)(39,41);(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,28,35,34)(17,36,33,37)(21,38,39,27)(22,41,42,32)(23,43,40,44)Polycyclic presentation:
2^9|b^2=fi,c^2=gh,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366988);