Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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)(13,14,18,22)(15,16,20,19)(17,25,26,21)(23,27,28,24)(29,30,34,37)(31,38,35,32)(33,36,42,44)(39,41,43,40);(6,9)(10,12)(15,20)(16,19)(23,28)(24,27)(29,31,34,35)(30,32,37,38)(33,39,42,43)(36,40,44,41);(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,19,22,16)(17,23,26,28)(21,27,25,24)(29,31)(30,32)(33,39)(34,35)(36,40)(37,38)(41,44)(42,43);(1,2)(3,4)(13,16,18,19)(14,20,22,15)(17,24,26,27)(21,28,25,23)(29,32)(30,35)(31,37)(33,41)(34,38)(36,39)(40,42)(43,44);(5,7,8,11)(6,10,9,12)(13,17,18,26)(14,21,22,25)(15,23,20,28)(16,24,19,27)(29,33,34,42)(30,36,37,44)(31,39,35,43)(32,40,38,41)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=fg,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2671095);