Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33531 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,13)(9,22)(10,20)(11,15)(12,18)(14,25)(17,33)(19,34)(21,24)(23,28)(26,35)(27,31)(29,32)(30,36);(5,7)(6,12)(8,17)(9,23)(10,27)(11,19)(13,28)(14,31)(15,29)(16,22)(18,25)(20,30)(21,32)(24,33)(26,36)(34,35);(5,6)(7,12)(8,20)(9,24)(10,13)(11,15)(14,26)(16,32)(17,30)(18,34)(19,29)(21,22)(23,33)(25,35)(27,28)(31,36);(5,8,11,10)(6,13,15,20)(7,17,19,27)(9,21,26,25)(12,28,29,30)(14,22,24,35)(16,33,34,31)(18,23,32,36)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,f^b=fi,h^b=hi,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249864, #72249868, #72249882, … |
| maximal normal subgroup of | #72249864, #72249868, #72249882, … |
| maximal quotient of | #72264995, #72280466, #72285321, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400729);