Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | |
1840 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)(13,14,19,25)(15,27,31,21)(16,22,17,23)(18,24,20,26)(28,40,33,36)(29,43,44,37)(30,42,35,38)(32,39,34,41);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,31)(14,21,25,27)(16,28,17,33)(18,30,20,35)(22,36,23,40)(24,38,26,42)(29,32,44,34)(37,39,43,41);(2,4)(13,16)(14,22)(15,28)(17,19)(18,34)(20,32)(21,36)(23,25)(24,41)(26,39)(27,40)(29,30)(31,33)(35,44)(37,38)(42,43);(2,4)(5,8,9,7)(6,10,11,12)(13,17,19,16)(14,23,25,22)(15,29,31,44)(18,34,20,32)(21,37,27,43)(24,41,26,39)(28,35,33,30)(36,42,40,38);(6,11)(10,12)(13,18,19,20)(14,24,25,26)(15,30,31,35)(16,32,17,34)(21,38,27,42)(22,39,23,41)(28,44,33,29)(36,43,40,37)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424737);