Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×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 | |
5510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(10,16)(11,17)(13,14)(18,19)(20,21)(23,24);(1,2,3,4)(5,6)(7,8)(9,10,14,18,15,19,13,16)(11,12,17,24,20,22,21,23);(6,8)(9,11,13,21,15,20,14,17)(10,12,16,23,19,22,18,24);(6,8)(9,12)(10,17)(11,16)(13,23)(14,24)(15,22)(18,20)(19,21);(11,20)(12,22)(17,21)(23,24)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237254, #72237256, #72239652, … |
| maximal normal subgroup of | #72237254, #72237256, #72239652, … |
| maximal quotient of | #72237269, #72264561, #72264792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350458);