Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220210 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,12,17,10,25)(7,15,9,13,19,29,24,28)(8,20,22,31,23,14,26,32)(18,33,34,27,35,30,21,36);(2,4)(5,7,12,19)(6,13,17,28)(8,18,23,35)(9,10,24,11)(14,27,20,36)(15,16,29,25)(21,22,34,26)(30,31,33,32);(1,2,3,4)(5,8,12,23)(6,14,17,20)(7,18,19,35)(9,21,24,34)(10,22,11,26)(13,27,28,36)(15,30,29,33)(16,31,25,32);(5,9,10,19,12,24,11,7)(6,15,16,28,17,29,25,13)(8,21,22,35,23,34,26,18)(14,30,31,36,20,33,32,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293212);