Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
4519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,12)(11,15,16,14)(18,22)(20,23);(6,8)(10,12)(14,15)(17,18,21,22)(19,20,24,23);(5,6)(7,8)(10,12)(14,15)(18,22)(20,23);(1,2)(3,4)(9,11,13,16)(10,14,12,15)(17,19,21,24)(18,23,22,20);(9,12,13,10)(11,14,16,15)(17,20,21,23)(18,19,22,24)Polycyclic presentation:
2^9|a^2=g,b^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232457, #72232484, #72232500, … |
| maximal normal subgroup of | #72232457, #72232484, #72232500, … |
| maximal quotient of | #72278148, #72279047, C2×#3858151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3765347);