Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,18,19)(15,16,21,24)(17,25,27,20)(22,26,28,23)(29,30,34,37)(31,32,39,35)(33,41,43,36)(38,42,44,40);(6,9)(10,12)(14,19)(15,21)(20,25)(22,28)(29,31,34,39)(30,35,37,32)(33,38,43,44)(36,42,41,40);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,22)(18,21)(19,24)(20,23)(25,26)(27,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,42)(40,41)(43,44);(1,2)(3,4)(13,16)(14,15)(17,26)(18,24)(19,21)(20,28)(22,25)(23,27)(29,32,34,35)(30,31,37,39)(33,42,43,40)(36,44,41,38);(5,7)(6,10)(8,11)(9,12)(13,17,18,27)(14,20,19,25)(15,22,21,28)(16,23,24,26)(29,33,34,43)(30,36,37,41)(31,38,39,44)(32,40,35,42)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4769817);