Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1928 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)(6,9)(7,11)(13,14)(15,17)(16,24)(18,19)(20,23)(21,25)(22,28)(26,27)(30,32)(33,41)(35,37)(38,44);(7,11)(10,12)(14,19)(16,25)(17,20)(22,26)(29,30,34,32)(31,37,39,35)(33,40,41,36)(38,42,44,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,19)(18,23)(21,27)(22,25)(24,28)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,21)(15,22)(17,27)(18,25)(19,24)(20,28)(23,26)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(13,17,18,20)(14,15,19,23)(16,27,25,28)(21,22,24,26)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4407798);