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 | |
11036 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)(13,21)(18,22)(19,24)(20,23);(5,6)(7,8)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20);(1,2,3,4)(10,15)(12,16)(18,22)(20,23);(1,2)(3,4)(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20);(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)Polycyclic presentation:
2^9|b^2=i,c^2=h,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 | #72280094, #72302976, #72307827, … |
| maximal normal subgroup of | #72280094, #72302976, #72307827, … |
| maximal quotient of | #72280858, #72281360, #72315813 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7586382);