Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,10,17,11,15,27,16)(7,18,21,8,13,23,20,12)(9,24,25,31,26,14,30,32)(19,34,36,28,29,33,35,22);(1,2,4,3)(5,7)(6,12)(8,15)(9,19)(10,21)(11,13)(14,28)(16,23)(17,18)(20,27)(22,24)(25,36)(26,29)(30,35)(31,34)(32,33);(1,3,4,2)(5,8)(6,13)(7,15)(9,22)(10,23)(11,12)(14,29)(16,21)(17,20)(18,27)(19,24)(25,34)(26,28)(30,33)(31,36)(32,35);(5,9,10,25,11,26,27,30)(6,14,17,32,15,24,16,31)(7,19,21,36,13,29,20,35)(8,22,23,34,12,28,18,33);(6,15)(8,12)(14,24)(16,17)(18,23)(22,28)(31,32)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=g,f^2=i,g^2=i,b^a=bf,d^a=di,e^a=ei,f^a=fi,f^b=fi,e^c=ei164 conjugacy classes — view the full character table.
| maximal subgroup of | #72226779, #72226796, #72239482, … |
| maximal normal subgroup of | #72226779, #72226796, #72239482, … |
| maximal quotient of | #72285505, #72286485, #72318410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277291);