Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,22)(16,28)(17,23)(18,24)(19,33)(20,27)(21,31)(25,39)(26,36)(29,38)(30,43)(32,35)(34,44)(37,40)(41,42);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,25)(17,32)(18,27)(19,26)(21,34)(22,24)(23,36)(28,42)(29,44)(30,41)(31,40)(33,35)(37,38)(39,43);(6,10)(7,11)(13,16,17,29,18,30,19,31)(14,21,33,43,24,38,23,28)(15,25,32,44,27,41,26,40)(20,34,35,39,22,37,36,42);(7,11)(9,12)(13,15)(14,22)(16,25)(17,32)(18,27)(19,26)(20,24)(21,37)(23,35)(28,39)(29,44)(30,41)(31,40)(33,36)(34,38)(42,43);(13,17,18,19)(14,23,24,33)(15,26,27,32)(16,29,30,31)(20,35,22,36)(21,28,38,43)(25,40,41,44)(34,39,37,42)Polycyclic presentation:
2^9|a^2=fi,c^2=ghi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366715);