Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359901 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2342 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13874 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,13,6,14,7,12)(3,9,16,10,4,17,19,25)(5,11,21,26,20,28,22,27)(15,23,30,32,18,31,29,24);(1,3,7,16,6,4,8,19)(2,9,12,10,14,17,13,25)(5,15,21,29,20,18,22,30)(11,23,26,24,28,31,27,32);(1,4,6,3)(2,10)(5,18,20,15)(7,19,8,16)(9,13)(11,24)(12,17)(14,25)(21,30,22,29)(23,27)(26,31)(28,32);(1,5)(2,11)(3,15)(4,18)(6,20)(7,21)(8,22)(9,23)(10,24)(12,26)(13,27)(14,28)(16,29)(17,31)(19,30)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285951, #72286194, #72286218, … |
| maximal normal subgroup of | #72285951, #72286194, #72286218, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267563);