Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417686 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15267 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2250 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)(5,9)(6,21,24,10)(13,30)(16,25)(18,32)(19,29)(20,28)(23,27)(26,31);(1,3,6,8,7,14,24,15)(2,4,10,12,11,17,21,22)(5,18,19,31,20,25,23,13)(9,26,27,32,28,30,29,16);(1,4)(2,8)(3,10)(5,16)(6,22)(7,17)(9,25)(11,15)(12,24)(13,29)(14,21)(18,28)(19,30)(20,32)(23,26)(27,31);(1,5)(2,9)(3,13)(4,16)(6,23)(7,20)(8,25)(10,29)(11,28)(12,30)(14,31)(15,18)(17,32)(19,24)(21,27)(22,26)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,d^b=de,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72283686, #72318876, #72318894, … |
| maximal normal subgroup of | #72283686, #72318876, #72318894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326244);