Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388379 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14519 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1364 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)(3,11,13,7)(4,8,16,24)(5,10)(6,9)(12,20)(14,26,18,21)(15,22)(17,23,30,32)(19,25)(27,28)(29,31);(1,3)(2,7)(4,15)(5,18)(6,13)(8,22)(9,11)(10,26)(12,17)(14,19)(16,29)(20,23)(21,25)(24,31)(27,32)(28,30);(1,4)(2,8,9,24)(3,12,13,28)(5,17)(6,16)(7,20)(10,23,25,32)(11,27)(14,15,18,29)(19,30)(21,22)(26,31);(5,19)(7,11)(10,25)(14,18)(15,29)(17,30)(20,27)(23,32)Polycyclic presentation:
2^9|a^2=h,c^2=eh,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295575);