Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,24)(23,25)(26,28);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(14,18)(19,24)(20,25)(26,27);(7,11)(9,12)(13,15)(14,19)(16,21)(17,22)(18,24)(20,26)(23,28)(25,27);(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,28)(24,27)Polycyclic presentation:
2^9|a^2=g,b^2=ei,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72267144, #72267145, #72267146, … |
| maximal normal subgroup of | #72267144, #72267145, #72267146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308780);