Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
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 | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4998 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)(7,8)(9,10,11,12)(13,14)(15,23)(16,22)(17,21)(18,20)(19,24)(25,27)(26,28);(2,4)(13,15,18,26)(14,19,20,27)(16,25,17,24)(21,28,22,23);(6,8)(15,24)(16,17)(19,23)(21,22)(25,26)(27,28);(14,20)(19,27)(21,22)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314500, #72314501, #72314509, … |
| maximal normal subgroup of | #72314500, #72314501, #72314509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264708);