Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| 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 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(10,14)(11,18)(12,15)(16,19)(17,22)(20,23);(1,2,3,4)(5,6)(7,8)(9,10,15,22,13,17,12,14)(11,19,23,21,20,16,18,24);(6,8)(9,11,12,18,13,20,15,23)(10,16,14,21,17,19,22,24);(6,8)(10,17)(14,22)(16,19)(21,24);(11,20)(16,19)(18,23)(21,24)Polycyclic presentation:
2^9|b^2=fi,c^2=g,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=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249555, #72249620, #72280210, … |
| maximal normal subgroup of | #72249555, #72249620, #72280210, … |
| maximal quotient of | #72265102, #72268442, #72307955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350436);