Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,28,34)(16,21,31,37)(17,22,18,23)(20,32,35,27)(26,39,42,43)(29,36,30,38)(33,41,44,40);(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,26)(17,27)(18,32)(19,28)(21,33)(22,34)(23,25)(24,35)(29,40)(30,41)(31,42)(36,43)(37,44)(38,39);(2,4)(7,11)(9,12)(13,16)(14,21)(15,26)(17,29)(18,30)(19,31)(20,33)(22,36)(23,38)(24,37)(25,39)(27,40)(28,42)(32,41)(34,43)(35,44);(6,10)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,29,31,30)(20,34,35,25)(21,36,37,38)(26,40,42,41)(33,43,44,39);(13,17,19,18)(14,23,24,22)(15,27,28,32)(16,30,31,29)(20,25,35,34)(21,36,37,38)(26,41,42,40)(33,43,44,39)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340056);