Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417630 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15376 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,13,16,9)(4,18)(5,8,12,7)(10,22)(14,27)(15,25,24,17)(19,26)(20,23)(21,28)(29,31)(30,32);(1,3)(2,9)(4,19)(5,24)(6,16)(7,17)(8,25)(10,27)(11,13)(12,15)(14,22)(18,26)(20,29)(21,32)(23,31)(28,30);(1,4)(2,10)(3,14)(5,20)(6,23)(7,22)(8,21)(9,26)(11,28)(12,18)(13,29)(15,30)(16,32)(17,31)(19,25)(24,27);(1,5,8,2,6,12,7,11)(3,15,25,13,16,24,17,9)(4,20,21,10,23,18,22,28)(14,30,19,29,32,27,31,26)Polycyclic presentation:
2^9|a^2=e,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72279915, #72287286, #72287308, … |
| maximal normal subgroup of | #72279915, #72287286, #72287308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335140);