Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254491 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(4,14)(8,24)(9,23)(10,21)(13,30)(15,19)(16,17)(20,26)(22,32)(25,31)(28,29);(2,8,10,15)(3,11)(7,19,21,24)(9,25,26,32)(12,27)(13,28)(20,31,23,22)(29,30);(1,2)(3,7)(4,15)(5,10)(6,9)(8,16)(11,23)(12,21)(13,31)(14,25)(17,32)(18,26)(19,30)(20,27)(22,29)(24,28);(2,9)(3,12)(7,20)(8,25)(10,26)(11,27)(13,29)(15,32)(19,31)(21,23)(22,24)(28,30)Polycyclic presentation:
2^9|b^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,g^c=gh,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338782, #72341122 |
| maximal normal subgroup of | #72338782 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161687);