Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23996 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14064 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,18,19)(3,7,13,22,12,20,16,25)(4,8,15,24,14,23,17,26)(11,27,29,32,28,21,30,31);(1,3,5,16,6,12,18,13)(2,7,19,22,9,20,10,25)(4,11,17,29,14,28,15,30)(8,21,24,32,23,27,26,31);(7,20)(21,27)(22,25)(31,32);(1,4)(2,8,9,23)(3,11)(5,17)(6,14)(7,21,20,27)(10,26,19,24)(12,28)(13,30)(15,18)(16,29)(22,32,25,31)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, #72240448, #72240459, … |
| maximal normal subgroup of | #72237642, #72240448, #72240459, … |
| maximal quotient of | #72237664, #72256194, C2×#371022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278218);