Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1849 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)(7,12)(8,10)(9,11)(13,14,18,23)(15,24,28,20)(16,21,30,37)(17,31,19,22)(25,39,41,33)(26,35,27,34)(29,38,32,36)(40,44,42,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,28)(14,20,23,24)(16,25,30,41)(17,27,19,26)(21,33,37,39)(22,34,31,35)(29,40,32,42)(36,44,38,43);(2,4)(13,16)(14,21)(15,25)(17,32)(18,30)(19,29)(20,33)(22,38)(23,37)(24,39)(26,40)(27,42)(28,41)(31,36)(34,43)(35,44);(2,4)(5,8,9,7)(6,10,11,12)(15,26)(17,19)(20,34)(22,31)(24,35)(25,40)(27,28)(29,32)(33,43)(36,38)(39,44)(41,42);(6,11)(10,12)(13,17,18,19)(14,22,23,31)(15,27,28,26)(16,29,30,32)(20,34,24,35)(21,36,37,38)(25,40,41,42)(33,44,39,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424387);