Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
10973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,15,16)(11,17,12,13)(14,18,23,24)(19,22,20,21);(5,6,7,8)(9,11,15,12)(10,13,16,17)(14,19,23,20)(18,21,24,22);(2,3)(10,16)(11,12)(18,24)(19,20);(1,3,4,2)(9,12,15,11)(10,17,16,13)(14,20,23,19)(18,22,24,21);(9,13,15,17)(10,11,16,12)(14,22,23,21)(18,20,24,19);(9,14,15,23)(10,18,16,24)(11,19,12,20)(13,21,17,22)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,e^2=i,f^2=i,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 | #72234472, #72234482, #72234843 |
| maximal normal subgroup of | #72234472, #72234482, #72234843 |
| maximal quotient of | #72284210, #72284323 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7602283);