Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417630 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15358 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1027 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)(3,12,15,23)(4,9)(5,14,22,16)(6,10)(7,24)(8,11)(13,25,30,29)(17,18,31,32)(19,26)(20,27)(21,28);(1,3,6,16,8,15,7,14)(2,5,10,23,11,22,24,12)(4,17,19,13,21,31,20,30)(9,25,26,32,28,29,27,18);(1,4,6,19,8,21,7,20)(2,9,10,26,11,28,24,27)(3,13,16,31,15,30,14,17)(5,18,23,25,22,32,12,29);(1,5)(2,3)(4,18)(6,23)(7,12)(8,22)(9,13)(10,16)(11,15)(14,24)(17,27)(19,25)(20,29)(21,32)(26,31)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338619, #72338624, #72338633, … |
| maximal normal subgroup of | #72338619, #72338624, #72338633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333944);