Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417738 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1970 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)(7,17)(8,14)(9,12)(10,15)(11,16)(13,20)(18,32)(19,28)(21,25)(22,30)(23,26)(24,29)(27,34)(31,35)(33,36);(6,12)(8,22)(9,10)(13,25)(14,29)(15,16)(17,28)(18,33)(19,20)(23,24)(31,32)(34,35);(2,4)(5,7)(6,13)(8,18)(9,20)(10,19)(11,21)(12,25)(14,27)(15,17)(16,28)(22,34)(23,33)(24,35)(26,32)(29,36)(30,31);(5,8)(6,14)(7,18)(9,23)(10,22)(11,24)(12,26)(13,27)(15,30)(16,29)(17,31)(19,34)(20,33)(21,35)(25,32)(28,36)Polycyclic presentation:
2^9|a^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326052);