Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13497 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2075 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,10,15)(7,16,19,12)(8,11,21,13)(9,14,24,31)(17,33,35,29)(18,28,20,25)(22,26,36,30)(23,27,34,32);(6,11)(7,17,19,35)(8,21)(9,22)(12,27,16,32)(13,15)(14,31)(18,23,20,34)(24,36)(25,29,28,33);(5,7,8,18,10,19,21,20)(6,12,13,28,15,16,11,25)(9,23,22,17,24,34,36,35)(14,32,30,33,31,27,26,29)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=di,f^b=fh,h^b=hi,e^c=ei,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239855, #72240848, #72240857, … |
| maximal normal subgroup of | #72239855, #72240848, #72240857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56152);