Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2676 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15263 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1643 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)(4,9)(5,10)(6,14)(7,11)(8,12)(15,29)(16,19)(17,24)(18,31)(20,32)(21,28)(22,27)(23,26)(25,30);(1,3)(2,4)(5,20)(6,16)(7,9)(8,17)(10,25)(11,24)(12,19)(13,14)(15,21)(18,26)(22,31)(23,30)(27,29)(28,32);(1,4)(2,9)(3,14)(5,18)(6,13)(7,24)(8,19)(10,20)(11,17)(12,16)(15,28)(21,25)(22,32)(23,29)(26,30)(27,31);(1,5,6,21,8,22,7,23)(2,10,11,26,12,27,14,28)(3,15,9,20,17,30,16,31)(4,18,13,25,19,32,24,29)Polycyclic presentation:
2^9|d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72215938, #72219482, … |
| maximal normal subgroup of | #72213913, #72215938, #72219482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325666);