Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
3278 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)(6,9)(7,11)(14,17)(15,19)(22,25)(26,27);(5,6,8,9)(7,10,11,12)(14,17)(18,20)(21,22,24,25)(23,27,28,26);(7,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22,24,25)(23,26,28,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(24,25)(26,28);(5,6)(7,12)(8,9)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23,24,28)(22,26,25,27)Polycyclic presentation:
2^9|b^2=f,c^2=gh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2991731);