Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| 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×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4269 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)(15,25)(16,30)(17,24)(18,23)(19,22)(20,28)(21,32)(26,40)(27,36)(29,35)(31,37)(33,38)(34,41)(39,43)(42,44);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,26)(17,29)(18,28)(19,27)(21,34)(22,35)(23,36)(24,25)(30,44)(31,43)(32,42)(33,41)(37,40)(38,39);(7,11)(9,12)(13,16,19,33)(14,21,22,37)(15,26,27,41)(17,32,18,31)(20,34,35,40)(23,30,24,38)(25,39,36,44)(28,43,29,42);(6,10)(9,12)(15,27)(20,35)(25,36)(26,41)(28,29)(34,40)(39,44)(42,43);(14,22)(20,35)(21,37)(23,24)(25,36)(30,38)(34,40)(39,44)Polycyclic presentation:
2^9|c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340119);