Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1535 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)(5,6,9,13,10,14,22,28)(7,15,17,23,18,27,21,11)(8,12,19,25,20,26,33,36)(16,29,30,34,31,35,32,24);(5,7)(6,11)(8,16)(9,21)(10,18)(12,24)(13,27)(14,23)(15,28)(17,22)(19,32)(20,31)(25,35)(26,34)(29,36)(30,33);(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=e,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 | #72250313, #72250427, #72252634, … |
| maximal normal subgroup of | #72250313, #72250427, #72252634, … |
| maximal quotient of | C2×#400129 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307325);