Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418871 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2697 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15266 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1932 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)(3,12,8,4)(5,17)(6,10)(7,21)(9,18)(11,20)(13,29,24,32)(14,16,22,27)(15,30,25,31)(19,28)(23,26);(1,3)(2,4)(5,13)(6,8)(7,22)(9,25)(10,12)(11,27)(14,20)(15,26)(16,21)(17,32)(18,31)(19,24)(23,30)(28,29);(1,4)(2,8)(3,10)(5,15)(6,12)(7,16)(9,24)(11,22)(13,26)(14,21)(17,31)(18,32)(19,25)(20,27)(23,29)(28,30);(1,5)(2,9)(3,13)(4,15)(6,19)(7,23)(8,24)(10,26)(11,28)(12,25)(14,31)(16,29)(17,21)(18,20)(22,30)(27,32)Polycyclic presentation:
2^9|a^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319954, #72320009, #72320059, … |
| maximal normal subgroup of | #72319954, #72320009, #72320059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326067);