Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,13,14,20)(8,11)(9,12)(10,15,17,19)(16,18);(1,3,4,2)(5,6)(7,10,14,17)(8,11)(9,12)(13,19,20,15)(16,18);(5,7)(6,10,12,17)(8,15)(9,14)(11,13,18,20)(16,19);(8,16)(11,18)(13,20)(15,19)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,e^c=ei,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219640, #72219671, #72219672, … |
| maximal normal subgroup of | #72219640, #72219671, #72219672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275420);