Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3516 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3014 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)(7,10)(8,9)(11,12)(13,14,19,25)(15,22,29,35)(16,21,31,38)(17,24,18,23)(20,28,36,27)(26,37,41,44)(30,34,42,43)(32,40,33,39);(1,2,3,4)(13,15,18,28,19,29,17,27)(14,20,24,35,25,36,23,22)(16,30,33,41,31,42,32,26)(21,37,40,43,38,44,39,34);(6,9)(10,12)(13,16,19,31)(14,21,25,38)(15,26,29,41)(17,33,18,32)(20,34,36,43)(22,37,35,44)(23,40,24,39)(27,30,28,42);(13,15,19,29)(14,22,25,35)(16,26,31,41)(17,28,18,27)(20,24,36,23)(21,37,38,44)(30,33,42,32)(34,39,43,40);(5,7,8,11)(6,10,9,12)(16,31)(21,38)(26,41)(30,42)(32,33)(34,43)(37,44)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6362147);