Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2185 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)(5,6,10,15)(7,16,19,29)(8,20,22,13)(9,12,24,14)(11,23,27,18)(17,31,33,36)(21,28,25,30)(26,32,35,34);(5,7)(6,11)(8,17)(9,23)(10,19)(12,16)(13,26)(14,29)(15,27)(18,24)(20,35)(21,34)(22,33)(25,32)(28,31)(30,36);(2,4)(6,12)(8,21)(9,24)(11,16)(13,20)(14,15)(17,32)(18,23)(22,25)(27,29)(31,36)(33,34);(5,8)(6,13)(7,17)(9,25)(10,22)(11,26)(12,28)(14,30)(15,20)(16,31)(18,34)(19,33)(21,24)(23,32)(27,35)(29,36)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 361196);