Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187949 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #62820 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,20)(3,11)(4,15)(8,21,22,30)(9,18,24,31)(12,14)(13,28)(16,29)(19,23,32,25)(26,27);(1,2)(3,8)(4,7)(5,10)(6,9)(11,25)(12,22)(13,19)(14,23)(15,20)(16,18)(17,24)(21,26)(27,32)(28,30)(29,31);(3,12)(4,16)(7,18)(8,22)(11,14)(13,26)(15,29)(19,21)(20,31)(23,25)(27,28)(30,32);(1,3,6,14)(2,8,9,23)(4,13,16,28)(5,12,17,11)(7,19,18,30)(10,22,24,25)(15,27,29,26)(20,32,31,21)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72266861, #72266864, #72266974, … |
| maximal normal subgroup of | #72266861, #72266864, #72266974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95148);