Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #24425 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2106 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)(7,9)(8,10)(11,12)(13,14,19,25)(15,21,28,37)(16,22,31,39)(17,24,18,23)(20,32,35,27)(26,36,42,43)(29,34,41,44)(30,40,33,38);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,20,25,35)(16,29,31,41)(17,27,18,32)(21,24,37,23)(22,36,39,43)(26,30,42,33)(34,38,44,40);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,26)(17,32)(18,27)(19,28)(20,23)(22,36)(24,35)(25,37)(29,33)(30,41)(31,42)(34,38)(39,43)(40,44);(7,12)(9,11)(13,16,19,31)(14,22,25,39)(15,26,28,42)(17,30,18,33)(20,34,35,44)(21,36,37,43)(23,38,24,40)(27,29,32,41);(6,10)(9,11)(13,17,19,18)(14,23,25,24)(15,27,28,32)(16,30,31,33)(20,21,35,37)(22,38,39,40)(26,29,42,41)(34,36,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6962102);