Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395242 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
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 | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,19)(16,20)(17,18)(21,22)(23,24);(9,11,15,22,16,23,14,18)(10,17,12,24,20,21,19,13);(1,2,3,4)(9,12,15,20,16,19,14,10)(11,21,22,13,23,17,18,24);(6,8)(9,13,14,21,16,24,15,17)(10,18,19,23,20,22,12,11)Polycyclic presentation:
2^9|b^2=ei,c^2=fi,d^2=e,e^2=i,f^2=i,g^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,g^b=gi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72226928, #72226931, #72226937 |
| maximal normal subgroup of | #72226928, #72226931, #72226937 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305657);