Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #197339 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2635 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,11)(3,14,17,30)(5,23,21,28)(6,12,20,16)(7,13,22,9)(8,27,29,24)(10,15,19,25)(18,26,31,32);(1,3,15,2,8,28,32,27)(4,10,25,9,29,31,26,16)(5,24,22,17,14,11,20,23)(6,18,21,12,7,19,30,13);(4,20)(5,25)(9,24)(10,23)(11,16)(14,26)(17,31)(22,29);(1,4)(2,9)(3,10)(5,21)(6,20)(7,22)(8,29)(11,13)(12,24)(14,30)(15,25)(16,27)(17,19)(18,23)(26,32)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213216, #72213613, #72213628, … |
| maximal normal subgroup of | #72213216, #72213613, #72213628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104536);