Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| 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 | |
3468 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)(9,10,13,18,14,17,21,15)(11,19,20,23,12,16,22,24);(2,4)(10,15)(11,20)(12,22)(13,21)(17,18)(23,24);(5,6,7,8)(9,11,21,22,14,12,13,20)(10,16,15,23,17,19,18,24);(9,12,21,20,14,11,13,22)(10,16,15,23,17,19,18,24);(10,17)(15,18)(16,19)(23,24)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=hi,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=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72312900, #72313010 |
| maximal normal subgroup of | #72312900, #72313010 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372826);