Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C2×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14,19,24)(15,25,28,22)(16,21,29,37)(17,20,33,36)(18,34,26,23)(27,38,42,40)(30,43,32,35)(31,44,41,39);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,26,33,19,28,18,17)(14,20,34,22,24,36,23,25)(16,27,41,30,29,42,31,32)(21,35,44,40,37,43,39,38);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,16)(14,21)(15,27)(17,32)(18,31)(19,29)(20,35)(22,40)(23,39)(24,37)(25,38)(26,41)(28,42)(30,33)(34,44)(36,43);(1,3,4,2)(7,12)(10,11)(15,28)(16,29)(17,33)(20,36)(21,37)(22,25)(31,41)(39,44);(13,17,18,28,19,33,26,15)(14,22,23,20,24,25,34,36)(16,30,31,27,29,32,41,42)(21,38,39,43,37,40,44,35)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=hi,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511629);