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 | #2397 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1780 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)(8,12)(9,11)(13,14,19,25)(15,21,30,38)(16,23,17,22)(18,24,20,26)(27,39,31,35)(28,44,43,36)(29,37,34,42)(32,41,33,40);(1,2,4,3)(5,7,9,12)(6,10,11,8)(13,15,19,30)(14,21,25,38)(16,31,17,27)(18,29,20,34)(22,39,23,35)(24,37,26,42)(28,32,43,33)(36,40,44,41);(5,8,9,10)(6,7,11,12)(13,16,19,17)(14,22,25,23)(15,27,30,31)(18,33,20,32)(21,35,38,39)(24,41,26,40)(28,29,43,34)(36,37,44,42);(6,11)(7,12)(13,17,19,16)(14,23,25,22)(15,28,30,43)(18,33,20,32)(21,36,38,44)(24,41,26,40)(27,29,31,34)(35,37,39,42);(1,3,4,2)(13,18,19,20)(14,24,25,26)(15,29,30,34)(16,32,17,33)(21,37,38,42)(22,40,23,41)(27,28,31,43)(35,36,39,44)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=g,d^2=i,e^2=fi,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,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7449306);