Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10)(11,15)(12,13)(14,16)(17,18,20,24,21,25,26,23)(19,22,29,31,28,30,27,32);(5,6,7,8)(10,13)(14,15)(17,19,26,27,21,28,20,29)(18,22,23,32,25,30,24,31);(2,4)(6,8)(18,23)(19,27)(20,26)(22,30)(24,25)(28,29);(9,11,12,16)(10,14,13,15)(19,28)(22,30)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264192, #72264214, #72264289, … |
| maximal normal subgroup of | #72264192, #72264214, #72264289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269108);