Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,12)(7,16)(8,20)(9,10)(15,24)(17,32)(18,19)(21,26)(22,23)(25,28)(27,33)(35,36);(1,2,3,4)(5,6,11,15)(7,13,19,29)(8,21,23,26)(9,24,10,12)(14,20,30,22)(16,25,18,28)(17,33,32,35)(27,31,36,34);(5,7,11,19)(6,13,15,29)(8,17,23,32)(9,18,10,16)(12,25,24,28)(14,27,30,36)(20,34,22,31)(21,35,26,33);(2,4)(5,8)(6,14)(7,17)(9,20)(10,22)(11,23)(12,26)(13,27)(15,30)(16,31)(18,34)(19,32)(21,24)(25,33)(28,35)(29,36)Polycyclic presentation:
2^9|b^2=ef,c^2=i,e^2=i,f^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,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328249);