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 | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#424 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2470 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)(5,6,10,15,11,7,16,14)(8,12,21,28,20,17,30,27)(9,13,24,29,25,18,23,22)(19,26,35,36,33,31,34,32);(5,7)(6,11)(8,17)(9,22)(10,14)(12,20)(13,24)(15,16)(18,23)(19,32)(21,27)(25,29)(26,35)(28,30)(31,34)(33,36);(1,2,3,4)(5,8)(6,12)(7,17)(9,19)(10,21)(11,20)(13,26)(14,27)(15,28)(16,30)(18,31)(22,32)(23,34)(24,35)(25,33)(29,36);(5,9)(6,13)(7,18)(8,19)(10,24)(11,25)(12,26)(14,22)(15,29)(16,23)(17,31)(20,33)(21,35)(27,32)(28,36)(30,34);(8,20)(12,17)(19,33)(21,30)(26,31)(27,28)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=h,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239492, #72250241, #72250290, … |
| maximal normal subgroup of | #72239492, #72250241, #72250290, … |
| maximal quotient of | #72318398, C2×#6380164, #72320111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287360);