Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476143 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2819 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17105 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,4)(5,17)(7,11,22,26)(8,13)(9,24)(12,27,18,31)(14,16)(15,28,19,32)(20,30)(21,25)(23,29);(1,3)(2,8,10,4)(5,18,20,12)(6,13)(7,14)(9,15)(11,23,26,16)(17,27)(19,25)(21,32,24,28)(22,29)(30,31);(1,4,6,8)(2,3)(5,19,20,15)(7,16,22,23)(9,12)(10,13)(11,14)(17,28)(18,25)(21,31,24,27)(26,29)(30,32);(1,5,6,20)(2,9)(3,12)(4,15,8,19)(7,21,22,24)(10,25)(11,17)(13,18)(14,28)(16,27,23,31)(26,30)(29,32)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^c=gi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72319935, #72320000, #72320059, … |
| maximal normal subgroup of | #72319935, #72320000, #72320059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384401);