Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(1,3,4,2)(6,9)(10,11)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(6,10)(8,12)(9,11)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27);(6,9)(10,11)(15,23)(18,21)(22,28)(25,27);(6,9)(10,11)(14,20)(18,21)(19,26)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=dfi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304366, #72304395, #72304457, … |
| maximal normal subgroup of | #72304366, #72304395, #72304457, … |
| maximal quotient of | #72305108, #72306190, #72306193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6295105);