Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2107 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)(5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,26)(17,20)(18,21)(19,23)(24,27)(25,28);(5,7,8,12)(6,9,10,11)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(13,15,18,25)(14,19,21,27)(16,24,17,23)(20,28,26,22);(2,4)(7,12)(9,11)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258248, #72258258, #72258259, … |
| maximal normal subgroup of | #72258248, #72258258, #72258259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325108);