Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,17)(16,20)(18,24)(19,23)(21,22);(10,14)(11,18)(12,20)(15,23)(16,17)(19,21);(2,4)(12,20)(14,16)(18,21)(22,24);(6,8)(9,11)(10,15)(12,21)(13,19)(14,22)(16,24)(17,23)(18,20)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258239, #72258258 |
| maximal normal subgroup of | #72258239, #72258258 |
| maximal quotient of | #72267312, #72267314 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356677);