Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1784 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)(7,15,18,27)(8,12)(9,13)(10,14)(11,17,23,21)(16,29,31,35)(19,25)(20,26)(22,28)(24,30,34,32)(33,36);(5,7,9,21,10,18,22,17)(6,11,13,27,14,23,28,15)(8,16,19,32,20,31,33,30)(12,24,25,35,26,34,36,29);(11,23)(15,27)(24,34)(29,35);(2,4)(5,8,9,19,10,20,22,33)(6,12,13,25,14,26,28,36)(7,16,21,32,18,31,17,30)(11,24,27,35,23,34,15,29)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249969, #72252700, #72252980, … |
| maximal normal subgroup of | #72249969, #72252700, #72252980, … |
| maximal quotient of | C2×#401034 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308230);