Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219498 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3154 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,9)(8,21,25,31)(10,16)(11,17)(12,19)(13,24,32,35)(14,26)(15,28)(18,33,34,29)(20,27)(22,23,30,36);(1,2,4,3)(5,7,10,19,11,20,28,14)(6,12,15,9,17,26,16,27)(8,22,35,18,25,30,24,34)(13,23,21,29,32,36,31,33);(5,8,11,25)(6,13)(7,18)(9,23,27,36)(10,24,28,35)(12,29,26,33)(14,30)(15,31)(16,21)(17,32)(19,22)(20,34);(1,3,4,2)(5,9,10,26,11,27,28,12)(6,14,15,7,17,19,16,20)(8,23,24,33,25,36,35,29)(13,30,31,18,32,22,21,34)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72255889, #72255911, #72255916 |
| maximal normal subgroup of | #72255889, #72255911, #72255916 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276379);