Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3496 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,3)(7,11)(10,12)(15,19)(17,20);(1,2,4,3)(6,9)(10,12)(13,14,16,18)(15,17,19,20);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,19)(18,20);(6,9)(10,12);(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,11)(9,12)Polycyclic presentation:
2^9|b^2=h,c^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72233768, #72233769, #72234466 |
| maximal normal subgroup of | #72233768, #72233769, #72234466 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538148);