Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7606 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)(11,20)(12,18)(13,19)(14,17)(15,22)(16,21)(23,24);(2,4)(5,6)(7,8)(9,11)(10,15)(12,21)(13,22)(14,23)(16,17)(18,20)(19,24);(6,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,17)(18,24)(19,20);(10,17)(15,16)(18,19)(20,24);(9,12,14,13)(10,18,17,19)(11,21,23,22)(15,20,16,24)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72224731, #72226727, #72237265, … |
| maximal normal subgroup of | #72224731, #72226727, #72237265, … |
| maximal quotient of | #72224710, #72239464, #72239664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338952);