Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| 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 | |
1791 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,14,10,16,9,15)(7,17,20,12,21,28,18,27)(8,22,25,13,24,29,23,30)(19,33,32,36,34,26,35,31);(1,2)(3,4)(5,7)(6,12)(8,19)(9,18)(10,21)(11,20)(13,26)(14,28)(15,17)(16,27)(22,36)(23,35)(24,34)(25,32)(29,31)(30,33);(2,4)(7,18)(8,23)(9,11)(12,17)(13,29)(14,15)(19,34)(20,21)(22,30)(24,25)(27,28)(31,36);(5,8,10,24)(6,13,16,30)(7,19,21,34)(9,25,11,23)(12,26,27,33)(14,22,15,29)(17,31,28,36)(18,32,20,35)Polycyclic presentation:
2^9|a^2=ef,d^2=i,e^2=i,g^2=i,h^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361861);