Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417541 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1318 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)(7,15,17,11)(8,12)(9,22)(10,13)(14,24)(16,25,32,31)(18,26,23,30)(19,33,34,27)(20,28)(21,36)(29,35);(5,7)(6,11)(8,19)(9,23)(10,17)(12,27)(13,15)(14,30)(16,21)(18,24)(20,34)(22,26)(25,29)(28,33)(31,36)(32,35);(5,8)(6,12)(7,16,17,32)(9,21)(10,20)(11,25,15,31)(13,28)(14,29)(18,19,23,34)(22,36)(24,35)(26,27,30,33);(2,4)(8,20)(11,15)(12,28)(16,32)(19,34)(21,35)(26,30)(29,36)Polycyclic presentation:
2^9|a^2=f,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285177, #72285571, #72285668 |
| maximal normal subgroup of | #72285177, #72285571, #72285668 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331803);