Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,14,16,20)(8,12)(9,18)(10,13)(11,17,19,15);(2,4)(7,15,16,17)(11,14,19,20);(5,7)(6,11)(8,15)(9,17)(10,16)(12,20)(13,19)(14,18);(2,4)(5,6)(7,11)(8,18)(9,12)(10,13)(14,15)(16,19)(17,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72258256, #72262282, #72262301, … |
| maximal normal subgroup of | #72258256, #72262282, #72262301, … |
| maximal quotient of | #72259001, #72303874 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271701);