Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(18,22)(20,28)(23,29)(24,30)(25,32)(27,31);(9,10)(11,14)(12,13)(15,16)(17,18,20,22,21,24,28,30)(19,25,27,32,26,23,31,29);(1,2,3,4)(10,13)(14,16)(17,19)(18,23)(20,27)(21,26)(22,29)(24,25)(28,31)(30,32);(9,11,12,15)(10,14,13,16)(19,26)(23,25)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=gi,e^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237232, #72237235, #72237292, … |
| maximal normal subgroup of | #72237232, #72237235, #72237292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266177);