Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #24226 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,12)(8,20)(9,14)(10,15)(11,16)(13,22)(17,26)(18,27)(19,28)(21,35)(23,30)(24,31)(25,32)(29,33)(34,36);(1,2,3,4)(5,7)(6,12)(8,21)(9,17)(10,18)(11,19)(13,29)(14,26)(15,27)(16,28)(20,35)(22,33)(23,32)(24,34)(25,30)(31,36);(5,8,11,9)(6,13,16,30)(7,17,19,21)(10,22,24,23)(12,25,28,29)(14,15,20,31)(18,32,34,33)(26,36,35,27);(5,9,11,8)(6,14,16,20)(7,17,19,21)(10,22,24,23)(12,26,28,35)(13,15,30,31)(18,33,34,32)(25,36,29,27)Polycyclic presentation:
2^9|b^2=e,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72281832, #72281927, #72281983, … |
| maximal normal subgroup of | #72281832, #72281927, #72281983, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419137);