Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#553 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,27)(25,29)(28,32)(30,34)(31,33)(35,36);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,25,30)(22,26,29,34)(24,31,32,36)(27,33,28,35);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35);(7,15)(10,13)(14,20)(17,19)(23,30)(26,34)(31,36)(33,35);(6,12)(10,13)(11,18)(17,19)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72292848, #72292850, #72292860 |
| maximal normal subgroup of | #72292848, #72292850, #72292860 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293354);