Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2459 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14806 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1401 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)(3,13)(4,17)(5,20)(6,12)(7,11)(8,15)(9,16)(10,19)(14,28)(18,26)(21,27)(22,24)(23,30)(25,31)(29,32);(1,3,7,22,6,16,5,15)(2,8,20,9,12,24,11,13)(4,14,18,29,19,31,21,30)(10,23,26,25,17,32,27,28);(1,3)(2,9)(4,14)(5,15)(6,16)(7,22)(8,11)(10,25)(12,13)(17,28)(18,29)(19,31)(20,24)(21,30)(23,27)(26,32);(1,4,7,18,6,19,5,21)(2,10,20,26,12,17,11,27)(3,14,22,29,16,31,15,30)(8,23,9,25,24,32,13,28)Polycyclic presentation:
2^9|b^2=ef,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215931, #72216141, #72219482, … |
| maximal normal subgroup of | #72215931, #72216141, #72219482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307919);