Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3516 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)(9,10,13,18,14,19,12,17)(11,20,21,24,22,15,23,16);(5,6)(7,8)(9,10)(11,20)(12,18)(13,17)(14,19)(15,22)(16,21)(23,24);(2,4)(6,8)(9,11)(10,15)(12,23)(13,21)(14,22)(16,18)(17,24)(19,20);(2,4)(9,11)(10,16)(12,21)(13,23)(14,22)(15,18)(17,20)(19,24);(9,12,14,13)(10,17,19,18)(11,21,22,23)(15,16,20,24)Polycyclic presentation:
2^9|a^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72315415, #72315429, #72315447 |
| maximal normal subgroup of | #72315415, #72315429, #72315447 |
| maximal quotient of | #72315862 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408043);