Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459490 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2733 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16599 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,12,5,8)(3,9,14,23,15,25,21,22)(4,16,20,10,19,27,18,28)(13,29,17,32,31,26,30,24);(2,8,12,11)(9,23,25,22)(10,16,28,27)(24,29,32,26);(1,3,7,14,6,15,5,21)(2,9,8,22,12,25,11,23)(4,17,20,31,19,30,18,13)(10,26,16,32,28,29,27,24);(1,4,6,19)(2,10,12,28)(3,13)(5,18,7,20)(8,16,11,27)(9,24)(14,17)(15,31)(21,30)(22,26)(23,29)(25,32)Polycyclic presentation:
2^9|a^2=efg,b^2=g,c^2=ef,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72280281, #72339678, #72339687, … |
| maximal normal subgroup of | #72280281, #72339678, #72339687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375211);