Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14207 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1403 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)(3,12)(4,16,19,9)(5,6)(7,11)(8,14)(10,22)(13,27,29,32)(15,25)(17,21,26,18)(20,24)(23,28,31,30);(1,3,6,20,7,15,22,14)(2,8,5,12,11,24,10,25)(4,13,18,30,19,29,21,28)(9,23,26,32,16,31,17,27);(1,4)(2,9)(3,13)(5,17)(6,21)(7,19)(8,23)(10,26)(11,16)(12,27)(14,30)(15,29)(18,22)(20,28)(24,31)(25,32);(1,5,6,11,7,10,22,2)(3,12,20,24,15,25,14,8)(4,17,21,16,19,26,18,9)(13,27,28,31,29,32,30,23)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=efh,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,h^c=hi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213889, #72219485, #72219491, … |
| maximal normal subgroup of | #72213889, #72219485, #72219491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285367);