Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #386374 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14427 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1053 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)(3,13,17,8)(4,18,20,28)(5,10)(6,12)(7,11)(9,21,27,22)(14,30)(15,25,16,24)(19,23)(26,31)(29,32);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,25)(11,13)(12,24)(14,21)(18,23)(20,32)(22,31)(27,30)(28,29);(1,4)(2,9,11,27)(3,14)(5,21)(6,22)(7,20)(8,23,13,29)(10,18,12,28)(15,19)(16,32)(17,31)(24,26,25,30);(2,10)(5,6)(8,24)(9,18)(11,12)(13,25)(15,16)(19,32)(21,22)(23,26)(27,28)(29,30)Polycyclic presentation:
2^9|a^2=h,c^2=efg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251777, #72251784, #72251795, … |
| maximal normal subgroup of | #72251777, #72251784, #72251795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293570);