Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(9,10,11,12)(13,14,19,24)(15,25,16,21)(17,22,18,23)(20,27,28,26);(1,2,3,4)(13,15,17,26,19,16,18,27)(14,20,23,25,24,28,22,21);(6,8)(13,16,19,15)(14,21,24,25)(17,26,18,27)(20,23,28,22);(13,17,19,18)(14,22,24,23)(15,26,16,27)(20,21,28,25)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^c=di,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72315686, #72315694, #72315745, … |
| maximal normal subgroup of | #72315686, #72315694, #72315745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264993);