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) = 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 | |
3187 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)(5,6,9,7)(8,10,12,11)(14,18)(16,19)(21,22)(23,26)(24,25)(27,28);(6,7)(8,12)(13,14)(15,19)(16,20)(17,18)(22,25)(23,27);(6,7)(10,11)(14,18)(15,20)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,17,20)(14,19,18,16)(21,22)(23,28)(24,25)(26,27);(5,8,9,12)(6,10,7,11)(13,16,17,19)(14,15,18,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=fh,c^2=fgh,d^2=gh,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, 2992750);