Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,16,20,29)(8,13)(9,23)(10,26)(11,15)(12,19,28,24)(14,17)(18,31,21,34)(22,30)(25,35)(27,32,36,33);(5,7,17,32,8,18,9,24,11,20,22,33,10,21,25,19)(6,12,23,34,26,36,14,29,15,28,35,31,13,27,30,16);(1,2,3,4);(5,8,11,10)(6,13,15,26)(7,18,20,21)(9,22,25,17)(12,27,28,36)(14,23,30,35)(16,31,29,34)(19,32,24,33)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=i,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72224815, #72224820, #72224821, … |
| maximal normal subgroup of | #72224815, #72224820, #72224821, … |
| maximal quotient of | #72225225, #72225248, #72225269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402330);