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 | |
4812 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)(9,10,14,11,16,18,15,19)(12,20,21,17,24,22,23,13);(2,4)(5,6)(7,8)(9,11)(10,14)(12,22)(13,23)(15,18)(16,19)(17,21)(20,24);(6,8)(9,12,16,24)(10,17,18,13)(11,20,19,22)(14,23,15,21);(6,8)(9,13)(10,12)(11,21)(14,20)(15,22)(16,17)(18,24)(19,23);(10,18)(11,19)(13,17)(20,22)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,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, #72305330, #72305332, … |
| maximal normal subgroup of | #72305326, #72305330, #72305332, … |
| maximal quotient of | #72305514, #72305613, #72305683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345084);