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 | |
11548 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)(6,8)(9,10)(11,19)(12,16)(13,14)(15,17)(18,24)(20,21)(22,23);(5,6,7,8)(9,11,15,12)(10,16,17,19)(13,22,18,20)(14,21,24,23);(2,4)(10,17)(11,12)(13,18)(21,23);(1,2)(3,4)(9,12,15,11)(10,16,17,19)(13,22,18,20)(14,23,24,21);(9,13,15,18)(10,14,17,24)(11,20,12,22)(16,23,19,21);(9,14)(10,18)(11,21)(12,23)(13,17)(15,24)(16,20)(19,22)Polycyclic presentation:
2^9|b^2=g,d^2=i,e^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 | #72234479, #72234484, #72234485, … |
| maximal normal subgroup of | #72234479, #72234484, #72234485, … |
| maximal quotient of | #72281703, #72282353, #72283184 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7585810);