Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2883 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,14)(8,11)(9,19)(10,13)(12,17)(15,16)(18,20);(1,2,3,4)(5,6,10,19,11,8,9,13)(7,14,17,15,18,20,16,12);(5,7)(6,12)(8,15)(9,16)(10,17)(11,18)(13,20)(14,19);(2,4)(5,8,11,6)(7,15,18,12)(9,13,10,19)(14,16,20,17)Polycyclic presentation:
2^9|b^2=ei,d^2=i,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,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226293, #72241171, #72241179, … |
| maximal normal subgroup of | #72226293, #72241171, #72241179, … |
| maximal quotient of | #72226233, C2×#417804 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325000);