Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1903 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)(5,6,10,8)(7,11,9,12)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(29,30)(31,38)(32,36)(33,41)(34,35)(37,42)(39,40)(43,44);(5,7,10,9)(6,11,8,12)(15,22)(17,25)(19,21)(20,24)(30,35)(31,36)(33,42)(40,44);(5,8,10,6)(7,11,9,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28)(29,31)(30,32)(33,43)(34,36)(35,38)(37,44)(39,42)(40,41);(1,2)(3,4)(5,9,10,7)(6,11,8,12)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,32)(30,36)(31,35)(33,44)(34,38)(37,39)(40,42)(41,43);(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33)(30,37)(31,39)(32,40)(34,42)(35,41)(36,43)(38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=gh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4052482);