Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #45136 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2008 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)(7,9)(8,10)(11,12)(13,14,20,25)(15,26,18,34)(16,22,31,40)(17,23,19,35)(21,32,24,28)(27,41,30,44)(29,37,33,39)(36,42,38,43);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,32,20,18,19,28)(14,21,35,34,25,24,23,26)(16,27,33,42,31,30,29,43)(22,36,37,44,40,38,39,41);(6,10)(7,11)(13,16)(14,22)(15,27)(17,33)(18,30)(19,29)(20,31)(21,36)(23,39)(24,38)(25,40)(26,41)(28,43)(32,42)(34,44)(35,37);(1,2,3,4)(6,10)(9,12)(13,17,20,19)(14,23,25,35)(15,28,18,32)(16,29,31,33)(21,34,24,26)(22,37,40,39)(27,42,30,43)(36,41,38,44);(13,18,19,32,20,15,17,28)(14,24,23,34,25,21,35,26)(16,30,29,42,31,27,33,43)(22,38,39,44,40,36,37,41)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=h,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518972);