Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7963 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(5,6)(7,8)(9,12)(10,11)(14,17)(18,19);(1,2,3,4)(5,7,9,11)(6,10,12,8);(5,8)(6,11)(7,12)(9,10)(13,14)(15,19)(16,17)(18,20);(5,7,9,11)(6,10,12,8)(13,15,16,20)(14,18,17,19);(5,8)(6,7)(9,10)(11,12);(5,6)(7,10)(8,11)(9,12)Polycyclic presentation:
2^9|b^2=g,d^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72234910, #72234911, #72234932, … |
| maximal normal subgroup of | #72234910, #72234911, #72234932, … |
| maximal quotient of | C2×#7694927 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7602123);