Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,22)(20,23)(21,24);(9,11)(10,15)(12,21)(13,22)(14,23)(16,19)(17,20)(18,24);(2,4)(9,10)(11,20)(12,17)(13,16)(14,18)(15,22)(19,23)(21,24);(9,12,14,13)(10,16,18,17)(11,21,23,22)(15,19,24,20)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239263, #72239267, #72239274, … |
| maximal normal subgroup of | #72239263, #72239267, #72239274, … |
| maximal quotient of | #72239458, #72239460, #72240283 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264769);