Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,9)(10,12)(13,14,19,32,18,22,17,23)(15,24,31,20,27,37,26,35)(16,28,33,36,29,21,30,43)(25,39,44,42,40,38,41,34);(2,3)(6,9)(10,11)(13,15,18,27)(14,20,22,35)(16,25,29,40)(17,31,19,26)(21,34,28,42)(23,37,32,24)(30,44,33,41)(36,39,43,38);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,16,18,29)(14,21,22,28)(15,25,27,40)(17,30,19,33)(20,34,35,42)(23,36,32,43)(24,38,37,39)(26,41,31,44);(1,3,4,2)(16,29)(21,28)(25,40)(30,33)(34,42)(36,43)(38,39)(41,44);(14,22)(20,35)(21,28)(23,32)(24,37)(34,42)(36,43)(38,39)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=gh,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373322);