Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10,15,11)(12,22,23,24)(13,17,19,20)(14,21,16,18);(5,6)(7,8)(9,11,14,21,15,10,16,18)(12,19,20,24,23,13,17,22);(2,4)(6,8)(9,12,14,20,15,23,16,17)(10,13,18,22,11,19,21,24);(6,8)(9,13)(10,17)(11,20)(12,18)(14,22)(15,19)(16,24)(21,23);(12,23)(13,19)(17,20)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308686, #72308796, #72308910, … |
| maximal normal subgroup of | #72308686, #72308796, #72308910, … |
| maximal quotient of | #72315776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346194);