Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| 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 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,12)(8,21)(9,10)(13,26)(14,22)(15,16)(17,28)(19,25)(27,31)(29,30)(32,36)(33,35);(5,6)(7,17)(8,22)(9,16)(10,12)(11,15)(13,19)(14,23)(18,32)(20,26)(21,29)(24,30)(25,28)(27,34)(31,33)(35,36);(2,4)(5,7)(6,13)(8,18)(9,19)(10,25)(11,20)(12,17)(14,27)(15,28)(16,26)(21,34)(22,31)(23,33)(24,35)(29,36)(30,32);(5,8,9,23,11,21,10,24)(6,14,12,22,15,30,16,29)(7,18,19,33,20,34,25,35)(13,27,17,31,28,32,26,36)Polycyclic presentation:
2^9|d^2=e,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,h^b=hi,g^c=gi,h^c=hi,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, 325680);