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 | |
3598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22)(23,28)(24,25)(26,27);(2,4)(6,7)(8,12)(14,16)(18,19)(21,23)(22,25)(24,26)(27,28);(1,2,3,4)(6,7)(10,11)(15,20)(18,19)(22,27)(23,24);(5,7,9,6)(8,10,12,11)(13,15)(14,18)(16,19)(17,20)(21,24,26,23)(22,25,27,28);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,18,20,19)(21,25)(22,24)(23,27)(26,28)Polycyclic presentation:
2^9|c^2=fhi,d^2=g,e^2=i,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.
| maximal subgroup of | #72300486, #72300508, #72300549, … |
| maximal normal subgroup of | #72300486, #72300508, #72300549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3145965);