Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2758 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)(7,11)(8,12)(9,14)(10,13)(15,31)(16,24)(17,25)(18,35)(19,27)(20,28)(21,30)(22,29)(23,32)(26,34)(33,36);(5,7)(6,11)(8,18,19,34)(9,17)(10,16)(12,26,27,35)(13,24)(14,25)(15,20,32,21)(22,33)(23,28,31,30)(29,36);(5,8)(6,12)(7,15)(9,21)(10,19)(11,23)(13,27)(14,30)(16,32)(17,34)(18,33)(20,22)(24,31)(25,35)(26,36)(28,29);(6,13)(11,24)(12,27)(14,29)(23,31)(25,36)(26,35)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=e,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,f^b=fi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72317312, #72317316, #72317332, … |
| maximal normal subgroup of | #72317312, #72317316, #72317332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264948);