Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,19,22,18,24,17,23)(15,20,32,35,26,34,27,36)(16,28,29,21,31,38,30,37)(25,39,41,33,40,44,42,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,27)(18,26)(19,32)(21,33)(22,35)(23,36)(24,34)(28,39)(29,41)(30,42)(31,40)(37,43)(38,44);(6,9)(10,12)(13,16)(14,21)(15,25)(17,29)(18,31)(19,30)(20,33)(22,28)(23,38)(24,37)(26,40)(27,41)(32,42)(34,43)(35,39)(36,44);(5,7,8,11)(6,10,9,12)(15,26)(20,34)(25,40)(27,32)(33,43)(35,36)(39,44)(41,42);(13,17,18,19)(14,22,24,23)(15,27,26,32)(16,29,31,30)(20,35,34,36)(21,28,37,38)(25,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,c^b=ch,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342689);