Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,11,6,9,18,24)(3,12,15,27,13,10,25,26)(5,7,17,21,14,19,29,31)(8,22,23,28,20,30,32,16);(2,7)(4,17)(8,12)(9,19)(10,20)(15,22)(18,29)(25,30);(2,8,7,12)(3,13)(4,17)(9,20,19,10)(11,27,21,23)(15,30)(16,28)(18,29)(22,25)(24,26,31,32);(2,9)(7,19)(8,20)(10,12)(11,24)(21,31)(23,32)(26,27)Polycyclic presentation:
2^9|a^2=g,c^2=e,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^b=gh,f^c=fi,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72233224, #72233309, #72233312, … |
| maximal normal subgroup of | #72233224, #72233309, #72233312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97518);