Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2265 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)(5,6,8,9)(7,10,11,12)(13,14,18,22,19,23,32,38)(15,24,17,30,26,37,31,20)(16,21,29,36,28,35,39,43)(25,34,27,41,40,44,42,33);(6,9)(10,12)(13,15,19,26)(14,20,23,30)(16,27,28,42)(17,18,31,32)(21,34,35,44)(22,37,38,24)(25,39,40,29)(33,43,41,36);(1,2,3,4)(13,16,18,29,19,28,32,39)(14,21,22,36,23,35,38,43)(15,25,17,27,26,40,31,42)(20,33,24,34,30,41,37,44);(5,7,8,11)(6,10,9,12)(13,17,18,26,19,31,32,15)(14,20,22,24,23,30,38,37)(16,27,29,40,28,42,39,25)(21,33,36,34,35,41,43,44);(16,28)(21,35)(25,40)(27,42)(29,39)(33,41)(34,44)(36,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367203);