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 | #45265 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ 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:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,25,29,20)(16,21,31,38)(17,22,19,24)(26,41,43,34)(27,42,44,35)(28,40,33,36)(30,37,32,39);(2,3)(6,9)(10,12)(13,15)(14,20)(16,26)(17,28)(18,29)(19,33)(21,34)(22,36)(23,25)(24,40)(27,30)(31,43)(32,44)(35,37)(38,41)(39,42);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16,18,31)(14,21,23,38)(15,26,29,43)(17,32,19,30)(20,34,25,41)(22,39,24,37)(27,28,44,33)(35,36,42,40);(1,3,4,2)(13,16)(14,21)(15,27)(17,30)(18,31)(19,32)(20,35)(22,37)(23,38)(24,39)(25,42)(26,33)(28,43)(29,44)(34,40)(36,41);(6,9)(10,12)(13,17,18,19)(14,22,23,24)(15,28,29,33)(16,30,31,32)(20,36,25,40)(21,37,38,39)(26,27,43,44)(34,35,41,42)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=g,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, 7467828);