Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #372790 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14100 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,11,6,3)(4,13,18,9,17,15,22,24)(5,10,21,14,19,25,20,26)(16,28,29,23,30,27,31,32);(1,3,7,11,8,12,6,2)(4,15,22,24,17,13,18,9)(5,14,20,25,19,26,21,10)(16,28,29,32,30,27,31,23);(1,4)(2,9)(3,13)(5,16)(6,22)(7,18)(8,17)(10,23)(11,24)(12,15)(14,27)(19,30)(20,31)(21,29)(25,32)(26,28);(1,5,6,21,8,19,7,20)(2,10,12,26,11,25,3,14)(4,16,22,29,17,30,18,31)(9,23,15,28,24,32,13,27)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72319403 |
| maximal normal subgroup of | #72319403 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279988);