Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4⋊C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
646 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)(5,6,10,15,9,16,19,35,11,17,24,36,27,14,32,30)(7,18,21,31,22,29,25,34,23,33,8,13,20,12,26,28);(5,7)(6,12)(8,24)(9,22)(10,21)(11,23)(13,30)(14,33)(15,28)(16,18)(17,29)(19,25)(20,27)(26,32)(31,35)(34,36);(1,2,3,4);(5,8)(6,13)(7,19)(9,26)(10,20)(11,21)(12,15)(14,34)(16,28)(17,31)(18,35)(22,24)(23,32)(25,27)(29,36)(30,33)Polycyclic presentation:
2^9|a^2=g,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=de,e^b=ei,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72319401, #72319557, D4×#2171 |
| maximal normal subgroup of | #72319401, #72319557, D4×#2171 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405022);