Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1454 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)(5,6,10,16,12,9,11,15)(7,17,21,19,22,30,20,13)(8,23,27,14,28,32,26,25)(18,34,35,31,36,29,24,33);(1,2,4,3)(5,7)(6,13)(8,24)(9,19)(10,21)(11,20)(12,22)(14,31)(15,30)(16,17)(18,27)(23,34)(25,33)(26,36)(28,35)(29,32);(5,8)(6,14)(7,18)(9,25)(10,27)(11,26)(12,28)(13,29)(15,23)(16,32)(17,33)(19,34)(20,24)(21,35)(22,36)(30,31);(1,3,4,2)(5,9)(6,12)(7,19)(8,25)(10,15)(11,16)(13,22)(14,28)(17,20)(18,34)(21,30)(23,27)(24,33)(26,32)(29,36)(31,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268710);