Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4874 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)(9,10,15,11,16,18,14,17)(12,21,20,24,23,19,22,13);(2,4)(5,6)(7,8)(9,11,16,17)(10,14,18,15)(12,21,23,19)(13,22,24,20);(6,8)(9,12)(10,13)(11,19)(14,20)(15,22)(16,23)(17,21)(18,24);(6,8)(9,13)(10,12)(11,20)(14,19)(15,21)(16,24)(17,22)(18,23);(9,14,16,15)(10,17,18,11)(12,20,23,22)(13,21,24,19)Polycyclic presentation:
2^9|a^2=fi,b^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72305326, #72305339, #72305344, … |
| maximal normal subgroup of | #72305326, #72305339, #72305344, … |
| maximal quotient of | #72306017, #72306018, #72306024, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345242);