Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3644 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)(9,10,15,17,14,18,13,11)(12,16,23,19,22,24,21,20);(2,4)(5,6)(7,8)(9,11)(10,15)(12,20)(13,18)(14,17)(16,23)(19,22)(21,24);(6,8)(9,12,14,22)(10,16,18,24)(11,19,17,20)(13,23,15,21);(9,10,14,18)(11,15,17,13)(12,16,22,24)(19,21,20,23)Polycyclic presentation:
2^9|a^2=ef,c^2=i,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72316992 |
| maximal normal subgroup of | #72316992 |
| maximal quotient of | #72317090 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302672);