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)(11,16)(12,18)(13,14)(15,17)(19,23)(20,22)(21,24);(5,6)(7,8)(9,11,15,20)(10,16,17,22)(12,19,21,13)(14,18,23,24);(2,4)(10,17)(11,20)(12,21)(14,23);(1,2)(3,4)(9,12)(10,18)(11,19)(13,20)(14,22)(15,21)(16,23)(17,24);(9,13,15,19)(10,14,17,23)(11,21,20,12)(16,24,22,18);(9,14)(10,19)(11,18)(12,22)(13,17)(15,23)(16,21)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72239419, #72249595, #72249598, … |
| maximal normal subgroup of | #72239419, #72249595, #72249598, … |
| maximal quotient of | #72264569, #72264734, #72264769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7590451);