Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3334 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)(9,10,13,18,14,17,22,23)(11,19,20,24,12,16,21,15);(6,7)(9,11)(10,15)(12,14)(13,21)(16,18)(17,24)(19,23)(20,22);(1,2,3,4)(5,6,8,7)(9,12,14,11)(10,16,17,19)(13,21,22,20)(15,23,24,18);(5,7,8,6)(11,12)(15,24)(16,19)(20,21);(10,17)(15,24)(16,19)(18,23)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72316421, #72316458, #72342362, … |
| maximal normal subgroup of | #72316421, #72316458, #72342362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372750);