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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2230 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)(5,6)(7,11)(8,9)(10,12)(13,14,19,25,20,26,18,24)(15,27,29,22,30,38,16,21)(17,23,34,41,35,42,33,40)(28,39,32,37,43,44,31,36);(2,4)(6,9)(10,11)(13,15)(14,21)(16,19)(17,32)(18,29)(20,30)(22,26)(23,39)(24,27)(25,38)(28,34)(31,35)(33,43)(36,41)(37,40)(42,44);(5,7,8,12)(6,10,9,11)(13,16,18,30,20,29,19,15)(14,22,24,27,26,21,25,38)(17,31,33,43,35,32,34,28)(23,37,40,39,42,36,41,44);(13,17,18,33,20,35,19,34)(14,23,24,40,26,42,25,41)(15,28,16,31,30,43,29,32)(21,36,38,44,22,37,27,39);(13,18,20,19)(14,24,26,25)(15,29,30,16)(17,33,35,34)(21,27,22,38)(23,40,42,41)(28,32,43,31)(36,39,37,44)Polycyclic presentation:
2^9|a^2=fgi,c^2=gh,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373392);