Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,15,9,14)(7,17,19,30,20,12,25,27)(8,21,22,13,24,28,23,29)(18,32,34,26,35,36,33,31);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,26)(14,27)(15,17)(16,30)(21,36)(22,33)(23,34)(24,35)(28,32)(29,31);(5,8)(6,13)(7,18)(9,22)(10,23)(11,24)(12,26)(14,28)(15,29)(16,21)(17,31)(19,33)(20,35)(25,34)(27,32)(30,36);(5,9,11,10)(6,14,15,16)(7,19,20,25)(8,22,24,23)(12,27,17,30)(13,28,29,21)(18,33,35,34)(26,32,31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^b=gi,f^c=fi,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72249921, #72250763, #72250782, … |
| maximal normal subgroup of | #72249921, #72250763, #72250782, … |
| maximal quotient of | #72249990, #72257685, #72263946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260558);