Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2811 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)(7,9)(8,15)(10,13)(11,14)(12,18)(16,17)(19,20);(5,7,10,14,11,13,9,6)(8,16,18,20,19,12,17,15);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,20)(14,16);(2,4)Polycyclic presentation:
2^9|b^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72252449, D4×#2008, #72252460, … |
| maximal normal subgroup of | #72252449, D4×#2008, #72252460, … |
| maximal quotient of | #72338199 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301480);