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×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1429 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)(8,12)(9,22)(10,14)(11,18)(13,23)(16,31)(17,26)(19,25)(20,28)(21,36)(24,34)(27,32)(29,35)(30,33);(5,7)(6,11)(8,16)(9,17)(10,19)(12,27)(13,25)(14,26)(15,22)(18,23)(20,33)(21,34)(24,28)(29,30)(31,36)(32,35);(5,8)(6,12)(7,16)(9,21)(10,20)(11,24)(13,29)(14,28)(15,30)(17,34)(18,32)(19,33)(22,36)(23,35)(25,31)(26,27);(2,4)(7,17,19,18)(8,20)(11,25,26,15)(12,28)(16,32,33,34)(21,35)(24,30,27,31)(29,36)Polycyclic presentation:
2^9|d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285507, #72285522, #72285668, … |
| maximal normal subgroup of | #72285507, #72285522, #72285668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334463);