Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1762 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,17,21,34)(8,22)(9,13)(10,26)(11,16)(12,19,30,18)(14,23)(15,28)(20,36,27,32)(24,29,35,33)(25,31);(5,7)(6,12,16,30)(8,18)(9,24)(10,27)(11,21)(13,17,31,34)(14,32,22,36)(15,33,26,29)(19,23)(20,28)(25,35);(1,2,3,4)(5,8,10,9,11,23,28,25)(6,13,15,22,16,31,26,14)(7,18,27,24,21,19,20,35)(12,17,33,36,30,34,29,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249918, #72249952, #72249974, … |
| maximal normal subgroup of | #72249918, #72249952, #72249974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54659);