Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3153 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1624 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)(5,6,11,16,12,18,10,17)(7,19,22,13,24,30,23,31)(8,25,9,14,28,33,27,15)(20,29,21,34,35,36,26,32);(5,7)(6,13)(8,26)(9,20)(10,23)(11,22)(12,24)(14,32)(15,34)(16,30)(17,19)(18,31)(21,28)(25,36)(27,35)(29,33);(2,4)(5,8,11,27,12,28,10,9)(6,14,16,25,18,15,17,33)(7,20,23,21,24,35,22,26)(13,29,19,34,31,36,30,32);(2,4)(5,9,10,28,12,27,11,8)(6,15,16,33,18,14,17,25)(7,21,22,20,24,26,23,35)(13,29,19,34,31,36,30,32)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293248);