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 | |
4766 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)(9,10,14,18,16,11,15,19)(12,21,24,17,20,23,22,13);(2,4)(5,6)(7,8)(9,11,16,10)(12,13,20,17)(14,18,15,19)(21,24,23,22);(6,8)(9,12,16,20)(10,17,11,13)(14,22,15,24)(18,21,19,23);(6,8)(9,13)(10,12)(11,20)(14,21)(15,23)(16,17)(18,24)(19,22);(9,14,16,15)(10,18,11,19)(12,22,20,24)(13,23,17,21)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^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, #72305327, #72305329, … |
| maximal normal subgroup of | #72305326, #72305327, #72305329, … |
| maximal quotient of | #72305911, #72306011, #72306027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345244);