Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2459 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14817 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1075 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)(3,7,13,22)(4,14)(5,17)(6,9)(8,16)(10,19)(11,24,18,20)(12,26,27,21)(15,23)(25,31,29,32)(28,30);(3,11,13,18)(7,20,22,24)(12,25,27,29)(21,31,26,32);(1,3)(2,7,9,22)(4,12)(5,18)(6,13)(8,21,14,26)(10,24,17,20)(11,19)(15,29)(16,27)(23,32,28,31)(25,30);(1,4)(2,8,9,14)(3,12)(5,15)(6,16)(7,21,22,26)(10,23,17,28)(11,25)(13,27)(18,29)(19,30)(20,31,24,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279915, #72285150, #72286050, … |
| maximal normal subgroup of | #72279915, #72285150, #72286050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308673);