Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1979 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)(7,10)(8,9)(11,12)(13,14)(15,27)(16,22)(17,23)(18,24)(19,25)(20,26)(21,31)(28,42)(29,40)(30,43)(32,44)(33,39)(34,41)(35,38)(36,37);(2,4)(13,15,19,31)(14,21,25,27)(16,32,29,35)(17,34,28,33)(18,30,20,36)(22,38,40,44)(23,39,42,41)(24,37,26,43);(6,9)(10,12)(13,16,19,29)(14,22,25,40)(15,28,31,17)(18,34,20,33)(21,23,27,42)(24,41,26,39)(30,35,36,32)(37,44,43,38);(13,17,19,28)(14,23,25,42)(15,29,31,16)(18,35,20,32)(21,22,27,40)(24,38,26,44)(30,34,36,33)(37,39,43,41);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,24,25,26)(15,30,31,36)(16,33,29,34)(17,35,28,32)(21,37,27,43)(22,39,40,41)(23,38,42,44)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361700);