Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2444 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)(5,6,10,14,11,15,9,22)(7,12,18,28,17,26,23,27)(8,13,20,31,21,32,24,30)(16,33,35,36,34,25,19,29);(2,4)(5,7,10,18,11,17,9,23)(6,12,14,28,15,26,22,27)(8,19,20,16,21,35,24,34)(13,29,31,33,32,36,30,25);(5,8)(6,13)(7,16)(9,24)(10,20)(11,21)(12,25)(14,31)(15,32)(17,34)(18,35)(19,23)(22,30)(26,33)(27,36)(28,29);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72322857 |
| maximal normal subgroup of | #72322857 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271617);