Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 16777216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1606 conjugacy classes of subgroups; only the 538 normal subgroups are shown.
This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 538 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(15,21)(16,23)(19,24)(20,25)(29,30,34,37)(31,38,35,32)(33,41,42,36)(39,40,43,44);(5,6,8,10)(7,9,12,11)(14,18)(19,24)(20,25)(26,28)(31,35)(32,38)(39,43)(40,44);(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,35)(30,32,37,38)(33,39,42,43)(36,40,41,44);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32)(30,35)(31,37)(33,40)(34,38)(36,43)(39,41)(42,44);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,42)(30,36,37,41)(31,39,35,43)(32,40,38,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371363);