Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #100171 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#2535 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14,17,21,18,20,16,25)(15,22,23,19,24,28,26,27);(6,9)(7,12)(10,11)(13,15)(14,19)(16,26)(17,23)(18,24)(20,27)(21,28)(22,25);(2,4)(14,20)(16,17)(22,28)(23,26)Polycyclic presentation:
2^9|a^2=g,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,d^c=di,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72240389, #72240390, #72240392, … |
| maximal normal subgroup of | #72240389, #72240390, #72240392, … |
| maximal quotient of | #72257698, #72257805 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22492);