Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1873 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)(7,10)(8,18,22,16)(9,17)(11,15)(12,21,29,20)(13,24)(14,27)(19,28,35,33)(23,26)(25,30)(31,34,32,36);(1,2)(3,4)(5,7,10,15,11,14,27,6)(8,12,21,18,22,29,20,16)(9,23,25,13,26,17,24,30)(19,33,34,31,35,28,36,32);(2,4)(5,8)(6,12)(7,16)(9,19)(10,20)(11,22)(13,28)(14,18)(15,29)(17,31)(21,27)(23,32)(24,34)(25,36)(26,35)(30,33);(5,9)(6,13)(7,17)(8,19)(10,25)(11,26)(12,28)(14,23)(15,30)(16,31)(18,32)(20,36)(21,34)(22,35)(24,27)(29,33)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267512);