Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #425083 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2576 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15954 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,10,21,25)(3,8,15,24,14,23,29,32)(4,16,18,22,9,5,19,20)(12,13,26,27,17,28,30,31);(1,3,6,14)(2,8)(4,17,9,12)(5,13)(7,15,21,29)(10,23)(11,24)(16,28)(18,30,19,26)(20,27)(22,31)(25,32);(1,4,6,9)(2,5,10,16)(3,12)(7,19,21,18)(8,13)(11,22,25,20)(14,17)(15,30)(23,28)(24,31)(26,29)(27,32);(1,5)(2,9)(3,13)(4,10)(6,16)(7,22)(8,12)(11,18)(14,28)(15,31)(17,23)(19,25)(20,21)(24,30)(26,32)(27,29)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338481, #72338485, #72338495 |
| maximal normal subgroup of | #72338481, #72338485, #72338495 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357156);