Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1451 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)(3,16,20,22)(4,21,24,26)(5,12)(6,29)(7,31)(8,14)(9,15)(10,18,25,11)(13,28)(17,27)(19,30,32,23);(1,3,6,11,7,32,5,26)(2,10,13,22,31,4,17,30)(8,19,28,21,9,20,27,18)(12,16,14,24,29,23,15,25);(1,4)(2,11)(3,17)(5,22)(6,30)(7,10)(8,25)(9,24)(12,19)(13,32)(14,21)(15,18)(16,28)(20,29)(23,27)(26,31);(1,5,7,6)(2,12,31,29)(3,18,32,21)(4,22,10,30)(8,27,9,28)(11,19,26,20)(13,14,17,15)(16,25,23,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72240203, #72240205, #72240213, … |
| maximal normal subgroup of | #72240203, #72240205, #72240213, … |
| maximal quotient of | #72238877, #72240341 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107600);