Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417661 |
| 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×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1708 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)(7,13,17,11)(8,12,20,30)(9,22)(10,14)(15,25)(16,26)(18,27,23,32)(19,36)(21,33,24,31)(28,35)(29,34);(5,7)(6,11)(8,19)(9,23)(10,17)(12,28)(13,14)(15,32)(16,21)(18,25)(20,35)(22,27)(24,34)(26,31)(29,33)(30,36);(2,4)(5,8,10,20)(6,12)(7,16,17,34)(9,24,25,21)(11,26)(13,29)(14,30)(15,33)(18,35,23,19)(22,31)(27,36)(28,32);(5,7)(6,13,14,11)(8,16)(9,18)(10,17)(12,29,30,26)(15,32,22,27)(19,21)(20,34)(23,25)(24,35)(28,31,36,33)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,d^2=e,g^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,f^b=fi,g^b=gi,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, 327724);