Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395242 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1890 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)(11,18)(12,23)(13,16)(14,17)(15,20)(19,21)(22,24);(2,4)(9,10,13,23,14,17,12,16)(11,19,21,18,22,15,20,24);(2,4)(9,10)(11,15)(12,16)(13,23)(14,17)(18,20)(19,22)(21,24);(6,8)(9,11,12,20,14,22,13,21)(10,15,16,18,17,19,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=efi,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216151 |
| maximal normal subgroup of | #72216151 |
| maximal quotient of | #72215990 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308224);