Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,11,17)(7,9,21,28)(8,22,25,32)(10,18,13,16)(12,15,27,20)(14,29,31,24)(19,26,33,36)(23,34,35,30);(5,7,12,18,13,15,28,17,11,21,27,16,10,20,9,6)(8,19,26,22,24,30,35,31,25,33,36,32,29,34,23,14);(1,3,4,2)(5,8,11,25)(6,14,17,31)(7,19,21,33)(9,23,28,35)(10,29,13,24)(12,26,27,36)(15,30,20,34)(16,32,18,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,h^a=hi,d^b=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250374, #72250383, #72250392, … |
| maximal normal subgroup of | #72250374, #72250383, #72250392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54386);