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 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3975 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)(7,16)(8,13)(9,25)(10,14)(11,15)(12,18)(17,28)(19,31)(20,27)(21,36)(22,32)(23,29)(24,30)(26,34)(33,35);(5,7)(6,12)(8,21)(9,20)(10,18)(11,19)(13,28)(14,31)(15,27)(16,25)(17,23)(22,35)(24,34)(26,29)(30,33)(32,36);(5,8,11,24)(6,13,15,30)(7,17,19,35)(9,23,10,22)(12,26,27,36)(14,32,25,29)(16,33,31,28)(18,21,20,34);(2,4)Polycyclic presentation:
2^9|c^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 | #72285452, #72285466, #72285641, … |
| maximal normal subgroup of | #72285452, #72285466, #72285641, … |
| maximal quotient of | #72316763, #72317028 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301237);