Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432256 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15453 |
| 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 | |
1392 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)(3,7,13,22)(4,8)(5,17)(6,10)(9,16)(11,19)(12,27,28,20)(14,31,30,23)(15,24)(18,25)(21,32,26,29);(1,3)(2,7,10,22)(4,12,15,28)(5,14)(6,13)(8,20)(9,26)(11,23,17,31)(16,29,18,32)(19,30)(21,25)(24,27);(1,4)(2,8)(3,12,13,28)(5,16)(6,15)(7,20,22,27)(9,17)(10,24)(11,25)(14,29,30,32)(18,19)(21,31,26,23);(1,4)(2,9)(3,12,13,28)(5,18)(6,15)(7,21,22,26)(8,17)(10,25)(11,24)(14,32,30,29)(16,19)(20,31,27,23)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, #72240435, #72240437, … |
| maximal normal subgroup of | #72237642, #72240435, #72240437, … |
| maximal quotient of | #72237652, C2×#432256, #72255610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339452);