Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16297 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1613 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)(3,12,14,7)(4,8)(5,11)(6,10)(9,15,26,19)(13,21,16,22)(17,24)(18,25)(20,27)(23,29,32,28)(30,31);(1,3)(2,7)(4,16)(5,15)(6,14)(8,22)(9,11)(10,12)(13,17)(18,29)(19,20)(21,24)(23,31)(25,32)(26,27)(28,30);(1,4)(2,8)(3,13,14,16)(5,18)(6,17)(7,21,12,22)(9,23,26,32)(10,24)(11,25)(15,28,19,29)(20,30)(27,31);(1,3)(2,9)(4,16)(5,19)(6,14)(7,11)(8,23)(10,26)(12,27)(13,17)(15,20)(18,28)(21,25)(22,31)(24,32)(29,30)Polycyclic presentation:
2^9|a^2=f,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72319592, #72319947, #72320059, … |
| maximal normal subgroup of | #72319592, #72319947, #72320059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364785);