Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434424 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15501 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9,20,26)(5,21)(6,22)(7,11)(8,14)(10,17)(13,28,16,30)(15,27)(18,24,29,23)(19,32,31,25);(1,3)(2,8)(4,16)(5,6)(7,15)(9,24)(10,27)(11,21)(12,22)(13,20)(14,17)(18,31)(19,29)(23,26)(25,28)(30,32);(1,4,17,31,7,20,6,19)(2,9,10,25,11,26,22,32)(3,13,5,18,15,16,14,29)(8,23,12,28,21,24,27,30);(1,5,17,15,7,14,6,3)(2,8,22,27,11,21,10,12)(4,18,31,16,20,29,19,13)(9,23,32,30,26,24,25,28)Polycyclic presentation:
2^9|a^2=g,c^2=ef,d^2=ei,e^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,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72339024, #72339031, #72339034, … |
| maximal normal subgroup of | #72339024, #72339031, #72339034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341621);