Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4812 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)(11,17)(12,19)(14,21)(16,22)(18,23)(20,24);(2,4)(5,6)(7,8)(9,10)(11,18)(12,16)(13,15)(14,20)(17,23)(19,22)(21,24);(6,8)(9,11,19,24,13,20,12,17)(10,14,22,23,15,18,16,21);(6,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24);(10,15)(11,20)(16,22)(17,24)Polycyclic presentation:
2^9|c^2=gi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249894, #72249904, #72249920, … |
| maximal normal subgroup of | #72249894, #72249904, #72249920, … |
| maximal quotient of | #72264376, #72265610 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349986);