Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15567 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23407 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,13,6,12)(3,14,9,4,16,18,25,19)(5,20,22,28,24,27,23,10)(15,17,30,26,29,32,21,31);(1,3,7,9,8,16,6,25)(2,4,11,18,13,19,12,14)(5,21,23,29,24,30,22,15)(10,26,27,17,28,31,20,32);(1,4)(2,9)(3,11)(5,17)(6,18)(7,14)(8,19)(10,21)(12,16)(13,25)(15,27)(20,29)(22,31)(23,26)(24,32)(28,30);(1,5)(2,10,13,28)(3,15,16,29)(4,17)(6,22)(7,23)(8,24)(9,21,25,30)(11,27,12,20)(14,26)(18,31)(19,32)Polycyclic presentation:
2^9|a^2=eg,b^2=f,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237064, #72237076, … |
| maximal normal subgroup of | #72237063, #72237064, #72237076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343900);