Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427241 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2471 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16051 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
937 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)(3,14,18,31)(4,10,5,11)(6,12)(7,23)(8,13)(9,16,26,17)(15,29)(19,25)(20,28,21,30)(22,24)(27,32);(1,3)(2,9,13,26)(4,19,5,22)(6,16)(7,17)(8,18)(10,27)(11,29)(12,14,23,31)(15,21,32,20)(24,30)(25,28);(1,4,6,21,8,5,7,20)(2,10,23,28,13,11,12,30)(3,15,16,19,18,32,17,22)(9,24,31,27,26,25,14,29);(1,5,6,20,8,4,7,21)(2,11,23,30,13,10,12,28)(3,15,16,19,18,32,17,22)(9,25,31,29,26,24,14,27)Polycyclic presentation:
2^9|a^2=egh,b^2=f,c^2=ef,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237069, #72237073, #72237094, … |
| maximal normal subgroup of | #72237069, #72237073, #72237094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359721);