Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,18,30)(8,19)(9,14)(10,13)(11,17,26,23)(12,22)(16,31,32,27)(20,25,34,36)(21,33)(24,29)(28,35);(5,7,9,23,10,18,24,17)(6,11,14,30,13,26,29,15)(8,20,21,16,22,34,35,32)(12,27,28,25,19,31,33,36);(2,4)(5,6)(7,11)(8,12)(9,14)(10,13)(15,17)(16,25)(18,26)(19,22)(20,27)(21,28)(23,30)(24,29)(31,34)(32,36)(33,35);(5,8,9,21,10,22,24,35)(6,12,14,28,13,19,29,33)(7,16,17,20,18,32,23,34)(11,25,15,27,26,36,30,31);(6,13)(11,26)(12,19)(14,29)(15,30)(25,36)(27,31)(28,33)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,d^b=df,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72264774, #72264797, #72264869, … |
| maximal normal subgroup of | #72264774, #72264797, #72264869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289112);