Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394463 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#397 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1882 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)(7,16,19,31)(8,13)(9,14)(10,25)(11,15)(12,18,27,20)(17,32,34,28)(21,35,33,26)(22,29)(23,36)(24,30);(5,7)(6,12)(8,21)(9,20)(10,18)(11,19)(13,28)(14,31)(15,27)(16,25)(17,23)(22,34)(24,33)(26,29)(30,32)(35,36);(5,8,9,22,11,24,10,23)(6,13,14,29,15,30,25,36)(7,17,18,33,19,34,20,21)(12,26,16,28,27,35,31,32);(2,4)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252917, #72256905, #72256910 |
| maximal normal subgroup of | #72252917, #72256905, #72256910 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302975);