Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455954 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16272 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1171 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)(4,19)(5,10)(6,23)(7,11)(8,12)(9,16)(14,25)(15,24)(17,26)(18,31)(20,30)(21,27)(22,28)(29,32);(1,3,6,16,8,18,7,17)(2,9,11,13,12,26,23,31)(4,20,22,14,5,15,21,32)(10,25,28,30,19,29,27,24);(1,4,6,21,8,5,7,22)(2,10,11,27,12,19,23,28)(3,14,16,20,18,32,17,15)(9,24,13,29,26,30,31,25);(1,5)(2,10)(3,15)(4,8)(6,21)(7,22)(9,25)(11,28)(12,19)(13,30)(14,17)(16,32)(18,20)(23,27)(24,31)(26,29)Polycyclic presentation:
2^9|b^2=ef,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237065, #72237066, #72237092, … |
| maximal normal subgroup of | #72237065, #72237066, #72237092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363150);