Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,15,24,25)(7,12,17,27,18,29,22,11)(8,13,23,32,21,31,20,30)(16,33,35,26,34,28,19,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,22)(10,18)(12,25)(13,26)(14,29)(15,27)(17,24)(19,23)(20,35)(21,34)(28,30)(31,36)(32,33);(2,4)(5,7)(6,12)(8,19,21,35)(9,17)(10,18)(11,25)(13,26,31,36)(14,27)(15,29)(16,20,34,23)(22,24)(28,30,33,32);(5,8)(6,13)(7,16)(9,23)(10,21)(11,26)(12,28)(14,32)(15,31)(17,35)(18,34)(19,22)(20,24)(25,30)(27,36)(29,33)Polycyclic presentation:
2^9|a^2=ef,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364802);