Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395586 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2411 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14725 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2142 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)(3,12,14,22)(4,15,17,8)(5,18)(6,21,23,10)(9,20)(13,27)(16,29)(19,25)(24,30)(26,28)(31,32);(1,3,6,8,7,14,23,15)(2,4,10,12,11,17,21,22)(5,13,19,24,20,28,32,29)(9,16,25,26,18,30,31,27);(1,4)(2,8)(3,10)(5,16)(6,22)(7,17)(9,24)(11,15)(12,23)(13,25)(14,21)(18,29)(19,27)(20,30)(26,32)(28,31);(1,5)(2,9)(3,13)(4,16)(6,19)(7,20)(8,24)(10,25)(11,18)(12,26)(14,28)(15,29)(17,30)(21,31)(22,27)(23,32)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338511, #72338520, #72338526 |
| maximal normal subgroup of | #72338511, #72338520, #72338526 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302782);