Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #389018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14533 |
| 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 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,8,5,21)(3,12,16,22,14,27,15,9)(4,17,20,10,19,25,18,23)(13,24,30,32,28,31,29,26);(2,8)(5,6)(9,22)(13,28)(15,16)(17,25)(18,20)(24,31);(1,3)(2,9,8,22)(4,13)(5,15)(6,16)(7,14)(10,24,23,31)(11,12,21,27)(17,26,25,32)(18,29)(19,28)(20,30);(1,4)(2,10)(3,13)(5,20)(6,18)(7,19)(8,23)(9,24)(11,17)(12,26)(14,28)(15,30)(16,29)(21,25)(22,31)(27,32)Polycyclic presentation:
2^9|a^2=h,c^2=fh,f^2=i,h^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,h^b=hi,e^c=ei,f^d=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72285150, #72286231, #72286267, … |
| maximal normal subgroup of | #72285150, #72286231, #72286267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296216);