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 | #35771 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1983 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)(5,6,8,10)(7,9,11,12)(13,14,19,24)(15,20,27,35)(16,21,30,38)(17,22,18,23)(25,33,42,44)(26,34,31,39)(28,36,41,43)(29,37,32,40);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,28)(17,26)(18,31)(19,27)(21,36)(22,34)(23,39)(24,35)(25,29)(30,41)(32,42)(33,37)(38,43)(40,44);(6,10)(7,11)(13,15)(14,20)(16,25)(17,31)(18,26)(19,27)(21,33)(22,39)(23,34)(24,35)(28,29)(30,42)(32,41)(36,37)(38,44)(40,43);(6,10)(9,12)(13,16,19,30)(14,21,24,38)(15,25,27,42)(17,29,18,32)(20,33,35,44)(22,37,23,40)(26,41,31,28)(34,43,39,36);(1,3,4,2)(13,17,19,18)(14,22,24,23)(15,26,27,31)(16,29,30,32)(20,34,35,39)(21,37,38,40)(25,41,42,28)(33,43,44,36)Polycyclic presentation:
2^9|a^2=gi,b^2=f,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, 7434938);