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 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1666 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)(5,6,10,25,11,15,20,14)(7,16,9,24,19,31,18,12)(8,13,27,29,23,30,22,36)(17,33,34,28,35,32,21,26);(5,7)(6,12)(8,17)(9,10)(11,19)(13,28)(14,31)(15,24)(16,25)(18,20)(21,22)(23,35)(26,30)(27,34)(29,32)(33,36);(2,4)(5,8,20,27,11,23,10,22)(6,13,25,36,15,30,14,29)(7,17,9,21,19,35,18,34)(12,26,31,33,24,28,16,32);(5,9,20,19,11,18,10,7)(6,12,25,31,15,24,14,16)(8,21,27,35,23,34,22,17)(13,26,36,33,30,28,29,32)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,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, 327692);