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)(15,25,27,22)(16,21,28,37)(17,20,32,36)(18,33,34,23)(26,38,41,40)(29,43,31,35)(30,44,42,39);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,18,17,19,27,34,32)(14,20,23,25,24,36,33,22)(16,26,30,31,28,41,42,29)(21,35,39,38,37,43,44,40);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,16)(14,21)(15,26)(17,31)(18,30)(19,28)(20,35)(22,40)(23,39)(24,37)(25,38)(27,41)(29,32)(33,44)(34,42)(36,43);(1,3,4,2)(7,12)(10,11)(15,27)(16,28)(17,32)(20,36)(21,37)(22,25)(30,42)(39,44);(13,17,18,27,19,32,34,15)(14,22,23,20,24,25,33,36)(16,29,30,26,28,31,42,41)(21,38,39,43,37,40,44,35)Polycyclic presentation:
2^9|a^2=gh,b^2=f,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,e^b=ei,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, 7511651);