Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,25)(16,20)(17,22)(18,33)(19,24)(21,28)(23,26)(27,39)(29,34)(30,35)(31,37)(32,43)(36,41)(38,40)(42,44);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,26,30,19,28,18,16)(14,20,33,21,24,35,23,25)(17,27,40,42,31,41,32,29)(22,34,43,36,37,44,38,39);(7,11)(9,12)(13,16,18,28,19,30,26,15)(14,21,23,20,24,25,33,35)(17,29,32,41,31,42,40,27)(22,36,38,34,37,39,43,44);(7,11)(9,12)(13,17)(14,22)(15,27)(16,29)(18,32)(19,31)(20,34)(21,36)(23,38)(24,37)(25,39)(26,40)(28,41)(30,42)(33,43)(35,44);(17,31)(22,37)(27,41)(29,42)(32,40)(34,44)(36,39)(38,43)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364553);