Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2804 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)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(2,4)(5,6)(7,8)(9,10)(11,19)(12,22)(13,16)(14,17)(15,21)(18,23)(20,24);(6,8)(9,11,13,20,14,21,12,23)(10,15,16,18,17,19,22,24);(6,8)(9,10)(11,15)(12,22)(13,16)(14,17)(18,20)(19,21)(23,24);(9,12,14,13)(10,16,17,22)(11,20,21,23)(15,24,19,18)Polycyclic presentation:
2^9|a^2=f,c^2=g,e^2=i,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 | #72249940, #72249945, #72249971, … |
| maximal normal subgroup of | #72249940, #72249945, #72249971, … |
| maximal quotient of | #72265605, #72307871 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350137);