Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2408 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1481 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)(7,17)(8,23)(9,15)(10,14)(11,16)(12,20)(13,25)(18,34)(19,33)(21,30)(22,29)(24,32)(26,31)(27,36)(28,35);(1,2,4,3)(5,7)(6,12)(8,19)(9,21)(10,20)(11,22)(13,28)(14,29)(15,17)(16,30)(18,25)(23,34)(24,36)(26,35)(27,31)(32,33);(5,8)(6,13)(7,18)(9,25)(10,24)(11,26)(12,27)(14,23)(15,31)(16,32)(17,33)(19,20)(21,35)(22,36)(28,29)(30,34);(1,3,4,2)(5,8)(6,13)(7,19)(9,24)(10,25)(11,26)(12,28)(14,31)(15,23)(16,32)(17,34)(18,20)(21,36)(22,35)(27,29)(30,33)Polycyclic presentation:
2^9|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=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307940);