Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417738 |
| 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×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1896 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)(6,11)(7,16)(9,25)(13,26)(14,15)(17,28)(18,33)(19,20)(21,30)(22,23)(27,36)(31,34);(1,2,3,4)(5,6,10,15)(7,17)(8,21,24,26)(9,14,25,11)(12,19)(13,23,30,22)(16,29)(18,27)(20,28)(31,32)(33,35)(34,36);(5,7)(6,12)(8,18)(9,19)(10,20)(11,17)(13,27)(14,28)(15,29)(16,25)(21,35)(22,31)(23,33)(24,34)(26,32)(30,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,26)(12,27)(14,21)(15,30)(16,31)(17,32)(19,33)(20,34)(22,25)(28,35)(29,36)Polycyclic presentation:
2^9|b^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328202);