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 | |
11249 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)(3,4)(5,6,7,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24);(5,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24);(2,4)(10,14)(11,18)(16,20)(19,24);(1,2)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72280094, #72302966, #72302988, … |
| maximal normal subgroup of | #72280094, #72302966, #72302988, … |
| maximal quotient of | #72281029, #72281051, #72283425 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7586780);