Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(5,6,7,8)(9,10)(11,13)(12,15)(14,16)(18,22)(23,31)(24,29)(25,32);(9,11)(10,12)(13,16)(14,15)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,30,31,27);(1,2,4,3)(9,12,14,13)(10,15,16,11)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25);(1,3,4,2)(9,13,14,12)(10,15,16,11)(18,22)(23,31)(24,29)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ef,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292694);