Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3424 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)(6,8)(9,10,15,19,16,21,14,20)(11,22,23,17,24,13,18,12);(2,4)(9,11)(10,17)(12,21)(13,20)(14,23)(15,18)(16,24)(19,22);(5,6,7,8)(9,12,15,22,16,17,14,13)(10,11,19,23,21,24,20,18);(9,13,15,12,16,22,14,17)(10,18,19,11,21,23,20,24);(9,14,16,15)(10,19,21,20)(11,23,24,18)(12,13,17,22)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=h,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307418, #72307511, #72315348 |
| maximal normal subgroup of | #72307418, #72307511, #72315348 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372837);