Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| 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×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2573 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)(7,13,15,17)(10,19,14,20);(2,4)(5,6)(7,14,15,10)(8,16)(9,12)(11,18)(13,19,17,20);(5,7)(6,10)(8,17)(9,15)(11,20)(12,14)(13,18)(16,19);(2,4)(8,18)(11,16)(13,17)(19,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232813, #72233091, #72233135, … |
| maximal normal subgroup of | #72232813, #72233091, #72233135, … |
| maximal quotient of | #72232796, #72233850 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271722);