Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4998 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)(11,12)(13,19)(14,20)(15,18)(16,17)(21,24)(22,23);(5,6)(7,8)(9,11)(10,16)(12,19)(13,21)(14,17)(15,22)(18,24)(20,23);(2,4)(6,8)(9,12,13,24,15,23,14,16)(10,17,20,22,18,21,19,11);(10,18)(12,23)(16,24)(19,20);(9,13,15,14)(10,19,18,20)(11,21,22,17)(12,24,23,16)Polycyclic presentation:
2^9|c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,g^b=gi,h^b=hi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72249525, #72249572, #72249575, … |
| maximal normal subgroup of | #72249525, #72249572, #72249575, … |
| maximal quotient of | #72264892, #72264900, #72311430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339160);