Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2666 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)(7,10)(8,9)(11,12)(13,14,20,26)(15,27,31,38)(16,32,33,41)(17,23,35,42)(18,25,19,24)(21,30,39,29)(22,34,40,28)(36,44,37,43);(1,2)(3,4)(6,9)(10,12)(13,15,18,29,20,31,19,30)(14,21,24,27,26,39,25,38)(16,17,34,36,33,35,28,37)(22,23,41,43,40,42,32,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,34,20,33,19,28)(14,22,24,41,26,40,25,32)(15,17,29,36,31,35,30,37)(21,23,27,43,39,42,38,44);(2,4)(13,17)(14,23)(15,28)(16,29)(18,36)(19,37)(20,35)(21,32)(22,27)(24,43)(25,44)(26,42)(30,33)(31,34)(38,40)(39,41);(16,33)(17,35)(22,40)(23,42)(28,34)(32,41)(36,37)(43,44)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348011);