Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417542 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2021 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(10,17)(11,15);(5,6)(7,10)(8,15)(9,12)(11,16)(13,18)(14,17)(19,20);(5,7)(6,10)(8,13)(9,14)(11,18)(12,17)(15,20)(16,19);(2,4)(5,8)(6,11,12,15)(7,13)(9,16)(10,18,17,20)(14,19)Polycyclic presentation:
2^9|d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225630, #72225637, #72225647, … |
| maximal normal subgroup of | #72225630, #72225637, #72225647, … |
| maximal quotient of | #72251441, #72265821, #72265822 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334456);