Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,22,18,20,17,21)(15,23,25,27,26,28,24,19);(13,15,17,24,18,26,16,25)(14,19,21,28,20,27,22,23);(6,9)(10,12);(5,7,8,11)(6,10,9,12)(14,20)(19,27)(21,22)(23,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72238865, #72238868, #72238871, … |
| maximal normal subgroup of | #72238865, #72238868, #72238871, … |
| maximal quotient of | #72238897 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261018);