Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,18,20)(15,16,17,19)(22,24)(23,27);(6,10)(7,12)(13,15,18,17)(14,19,20,16)(22,24)(26,28);(7,12)(8,11)(15,17)(16,19)(21,22,25,24)(23,28,27,26);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,16,18,19)(14,15,20,17)(21,23,25,27)(22,26,24,28);(5,8,9,11)(6,12,10,7)(13,17,18,15)(14,19,20,16)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5120709);