Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,22,25,21,24,30,32)(19,26,28,29,27,23,31,20);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,19)(18,23)(20,25)(21,27)(22,28)(24,26)(29,32)(30,31);(17,20,22,26,21,29,30,23)(18,19,25,28,24,27,32,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,24)(20,29)(23,26)(25,32);(18,24)(19,27)(25,32)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,d^a=di,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72306666, #72306674, #72306685, … |
| maximal normal subgroup of | #72306666, #72306674, #72306685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252893);