Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457092 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
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 | |
1622 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)(10,15)(12,13)(16,19)(17,18)(20,21)(23,24);(5,6,7,8)(9,10,13,17,14,18,12,15)(11,19,20,24,22,23,21,16);(2,4)(6,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,23)(18,24);(11,20,22,21)(16,19,24,23)Polycyclic presentation:
2^9|b^2=g,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213160, C2×#475764, #72214409, … |
| maximal normal subgroup of | #72213160, C2×#475764, #72214409, … |
| maximal quotient of | #72213145, #72213241, #72213348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382960);