Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419919 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
1243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(8,21)(9,10)(13,27)(14,22)(15,16)(17,29)(19,25)(26,30)(28,35)(31,36)(32,34);(1,2)(3,4)(5,6)(7,17)(8,22)(9,16)(10,12)(11,15)(13,19)(14,23)(18,28)(20,27)(21,26)(24,30)(25,29)(31,32)(33,36)(34,35);(2,4)(5,7)(6,13)(8,18)(9,19)(10,25)(11,20)(12,17)(14,28)(15,29)(16,27)(21,33)(22,35)(23,32)(24,34)(26,31)(30,36);(5,8,9,24,11,21,10,23)(6,14,16,22,15,30,12,26)(7,18,19,34,20,33,25,32)(13,28,27,35,29,36,17,31)Polycyclic presentation:
2^9|d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335068);