Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
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 | C23×C22⋊C4 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,15,18,28)(8,12)(9,14)(10,13)(11,17,24,21)(16,29,30,36)(19,25)(20,26)(22,27)(23,31,34,33)(32,35);(5,7,9,21,10,18,22,17)(6,11,14,28,13,24,27,15)(8,16,20,33,19,30,32,31)(12,23,26,36,25,34,35,29);(1,2,3,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,25)(14,26)(15,29)(17,31)(18,30)(21,33)(22,32)(24,34)(27,35)(28,36);(8,19)(12,25)(16,30)(20,32)(23,34)(26,35)(29,36)(31,33);(6,13)(11,24)(12,25)(14,27)(15,28)(23,34)(26,35)(29,36)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^b=fi,h^b=hi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72341714, #72341724, #72341753, … |
| maximal normal subgroup of | #72341714, #72341724, #72341753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279232);