Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| 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 | |
5220 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)(9,10)(11,12)(13,18)(14,19)(15,20)(16,17)(21,23)(22,24);(1,2,3,4)(5,6)(7,8)(9,11,13,17,15,22,14,21)(10,16,19,12,20,23,18,24);(6,8)(9,12,13,23,15,24,14,16)(10,17,19,22,20,21,18,11);(6,8)(9,10)(11,16)(12,17)(13,19)(14,18)(15,20)(21,24)(22,23);(9,13,15,14)(10,18,20,19)(11,17,22,21)(12,16,24,23)Polycyclic presentation:
2^9|b^2=f,c^2=gi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237259, #72237260, #72239652, … |
| maximal normal subgroup of | #72237259, #72237260, #72239652, … |
| maximal quotient of | #72237269, #72264552 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350468);