Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7493 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1390 |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C2×#127 | C23 |
| socle | C2 | C2×#1390 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,15,22)(4,8)(5,11,19,9)(6,10)(7,14,21,18)(13,26,28,32)(16,25,30,23)(17,24)(20,27,31,29);(1,3)(2,7,10,22)(4,13)(5,18)(6,14)(8,20,24,32)(9,21,11,12)(15,19)(16,29)(17,27)(23,31,25,26)(28,30);(1,4,6,17)(2,8,10,24)(3,13,14,27)(5,16,19,30)(7,20,22,32)(9,23,11,25)(12,26,21,31)(15,28,18,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fi,g^a=gh,h^a=hi,d^b=di,e^b=egi,g^b=gi,h^b=hi,e^d=ehi,g^d=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279362, #72279391, #72279489, … |
| maximal normal subgroup of | #72279362, #72279391, #72279489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59010);