Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480476 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1262 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)(7,15,18,29)(8,19)(9,13,23,27)(10,14)(11,17)(12,21)(16,30,32,28)(20,34,36,26)(22,25)(24,35)(31,33);(5,7,9,22)(6,11,13,15)(8,16,20,35)(10,18,23,17)(12,24,26,28)(14,25,27,29)(19,33,34,30)(21,32,36,31);(1,2,3,4);(5,8,10,21)(6,12,14,19)(7,16,18,32)(9,20,23,36)(11,24,25,33)(13,26,27,34)(15,28,29,30)(17,31,22,35)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312413, #72312415, #72312419, … |
| maximal normal subgroup of | #72312413, #72312415, #72312419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402371);