Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1714 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)(5,6,10,25,11,16,23,15)(7,17,24,14,19,31,9,12)(8,13,27,29,22,30,21,36)(18,33,20,28,34,32,35,26);(5,7)(6,12)(8,18)(9,23)(10,24)(11,19)(13,28)(14,16)(15,31)(17,25)(20,27)(21,35)(22,34)(26,30)(29,32)(33,36);(2,4)(5,8,10,21,11,22,23,27)(6,13,15,29,16,30,25,36)(7,18,9,20,19,34,24,35)(12,26,17,32,14,28,31,33);(5,9,23,7,11,24,10,19)(6,14,25,17,16,12,15,31)(8,20,27,18,22,35,21,34)(13,28,36,32,30,26,29,33)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,f^c=fi,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, 327690);