Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1714 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)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20)(21,27)(22,23)(24,26)(25,28)(29,30,34,35)(31,36,38,44)(32,41,37,33)(39,43,42,40);(5,6,9,10)(7,8,12,11)(14,16)(19,23)(20,24)(27,28)(30,35)(31,38)(32,37)(40,43);(6,10)(8,11)(13,15)(14,19)(16,23)(17,25)(18,22)(20,28)(21,26)(24,27)(29,31)(30,36)(32,42)(33,43)(34,38)(35,44)(37,39)(40,41);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,14)(15,19,22,23)(17,24,26,20)(21,27,25,28)(29,32,34,37)(30,33,35,41)(31,39,38,42)(36,40,44,43);(5,8,9,11)(6,7,10,12)(13,17,18,26)(14,20,16,24)(15,21,22,25)(19,27,23,28)(29,33)(30,37)(31,40)(32,35)(34,41)(36,42)(38,43)(39,44)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4990228);