Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14534 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1233 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)(3,14)(4,15,21,30)(5,22,25,11)(6,26)(7,12)(8,13)(9,16)(10,20,31,27)(17,29)(18,28)(19,23,32,24);(1,3,6,16,8,18,7,17)(2,9,12,14,13,29,26,28)(4,19,20,11,21,32,27,22)(5,10,23,30,25,31,24,15);(1,4,7,27,8,21,6,20)(2,10,12,30,13,31,26,15)(3,11,17,19,18,22,16,32)(5,14,23,29,25,28,24,9);(1,5)(2,11)(3,15)(4,14)(6,24)(7,23)(8,25)(9,20)(10,17)(12,19)(13,22)(16,31)(18,30)(21,28)(26,32)(27,29)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,g^b=gi,f^c=fi,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 | #72279915, #72284930, #72287267, … |
| maximal normal subgroup of | #72279915, #72284930, #72287267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296322);