Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(9,10,12,11)(13,14)(15,20)(16,21)(17,19)(18,22)(23,27)(24,26)(25,28);(5,6)(7,8)(9,11,12,10)(14,19)(16,17)(20,26)(21,22)(23,24)(27,28);(2,4)(6,8)(13,15,18,25)(14,20,22,28)(16,23,17,24)(19,26,21,27);(13,16)(15,23)(17,18)(19,21)(24,25)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72315795, #72315807, #72315809 |
| maximal normal subgroup of | #72315795, #72315807, #72315809 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271766);