Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480482 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,18,29)(8,19)(9,13,23,27)(10,14)(11,17)(12,20)(16,30,31,28)(21,34,35,26)(22,25)(24,36)(32,33);(5,7,9,22)(6,11,13,15)(8,16,21,36)(10,18,23,17)(12,24,26,28)(14,25,27,29)(19,33,34,30)(20,31,35,32);(2,4)(8,20)(12,19)(16,31)(21,35)(24,33)(26,34)(28,30)(32,36);(5,8,10,20)(6,12,14,19)(7,16,18,31)(9,21,23,35)(11,24,25,33)(13,26,27,34)(15,28,29,30)(17,32,22,36)Polycyclic presentation:
2^9|a^2=g,b^2=egi,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,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284021, #72285379, #72285425, … |
| maximal normal subgroup of | #72284021, #72285379, #72285425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401919);