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) = 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 | |
3198 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22,24,26)(23,27,28,25);(6,8)(10,11)(13,14)(15,19)(16,18)(17,20)(21,22,24,26)(23,25,28,27);(6,8)(7,12)(13,14,16,18)(15,17,20,19)(21,23,24,28)(22,25,26,27);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,25)(24,28)(26,27);(5,8,9,6)(7,10,12,11)(13,15,16,20)(14,17,18,19)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=fg,e^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.
| maximal subgroup of | #72263439, #72263447, #72263485 |
| maximal normal subgroup of | #72263439, #72263447, #72263485 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4056624);