Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4346 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)(8,21)(9,26)(10,14)(11,15)(12,19)(13,24)(17,34)(18,29)(20,28)(22,33)(23,32)(25,31)(27,35)(30,36);(5,7)(6,12)(8,22)(9,20)(10,19)(11,18)(13,30)(14,29)(15,28)(16,26)(17,23)(21,27)(24,35)(25,36)(31,34)(32,33);(5,8)(6,13)(7,17)(9,23)(10,24)(11,25)(12,27)(14,31)(15,32)(16,33)(18,35)(19,22)(20,36)(21,26)(28,34)(29,30);(7,18)(12,28)(16,29)(17,35)(19,20)(22,36)(27,34)(30,33)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226715, #72226729, #72237504, … |
| maximal normal subgroup of | #72226715, #72226729, #72237504, … |
| maximal quotient of | #72226704, #72249740, #72249745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264740);