Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459299 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2752 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16379 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,20,6,10,5,11)(3,12,14,27,15,8,21,23)(4,16,19,9,18,25,17,24)(13,28,29,22,30,32,31,26);(1,3,5,14,6,15,7,21)(2,8,11,23,10,12,20,27)(4,13,17,29,18,30,19,31)(9,22,16,32,24,26,25,28);(1,4)(2,9)(3,13)(5,19)(6,18)(7,17)(8,22)(10,24)(11,25)(12,26)(14,31)(15,30)(16,20)(21,29)(23,28)(27,32);(9,24)(16,25)(22,26)(28,32)Polycyclic presentation:
2^9|a^2=fg,b^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237064, #72237068, #72237069, … |
| maximal normal subgroup of | #72237064, #72237068, #72237069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368294);