Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18,20,24,21,25,28,23)(19,22,27,31,26,30,32,29);(1,2)(3,4)(5,7,8,6)(10,13)(14,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(2,4)(6,7)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(9,11,12,16)(10,14,13,15)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265653, #72265663, #72265684, … |
| maximal normal subgroup of | #72265653, #72265663, #72265684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323440);