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 | |
12114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24);(2,3)(10,11)(12,19)(16,18)(20,24);(1,3,4,2)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22);(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72234473, #72234475, #72234482, … |
| maximal normal subgroup of | #72234473, #72234475, #72234482, … |
| maximal quotient of | #72240162, #72261458, #72261470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581016);