Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394040 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4031 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,20)(18,28)(19,27)(21,31)(22,36)(23,29)(24,32)(25,30)(26,34)(33,35);(5,7)(6,12)(8,22)(9,19)(10,20)(11,21)(13,28)(14,17)(15,31)(16,27)(18,23)(24,35)(25,34)(26,29)(30,33)(32,36);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,29)(15,32)(16,30)(17,33)(19,34)(20,22)(21,35)(27,36)(28,31);(2,4)(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,25,24)(12,17,27,31)(13,29,30,32)(18,34,35,22)(26,33,36,28)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239541, #72255672, #72256257, … |
| maximal normal subgroup of | #72239541, #72255672, #72256257, … |
| maximal quotient of | #72239577, C2×#394042, #72317028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301238);