Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
3972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,14,13)(11,12,16,15)(18,23)(19,21);(6,8)(9,11,14,16)(10,15,13,12)(17,18,22,23)(19,24,21,20);(5,6)(7,8)(17,19,22,21)(18,24,23,20);(1,2)(3,4)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24);(9,13,14,10)(11,15,16,12)(17,21,22,19)(18,24,23,20)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72300333, #72300348, #72300451, … |
| maximal normal subgroup of | #72300333, #72300348, #72300451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1325716);