Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| 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×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3438 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)(9,10,12,16,13,15,21,23)(11,17,19,22,18,24,20,14);(6,7)(9,11)(10,14)(12,20)(13,18)(15,22)(16,24)(17,23)(19,21);(1,2,3,4)(5,6,8,7)(10,15)(11,18)(16,23)(19,20);(5,7,8,6)(11,18)(14,22)(17,24)(19,20);(10,15)(14,22)(16,23)(17,24)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=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72316421, #72316450, #72342362, … |
| maximal normal subgroup of | #72316421, #72316450, #72342362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372742);