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 | |
11084 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)(10,15)(13,20)(16,23)(18,24);(5,6)(7,8)(9,10,14,15)(11,16,19,23)(12,20,22,13)(17,18,21,24);(2,4)(10,15)(13,20)(16,23)(18,24);(1,2)(3,4)(9,11)(10,16)(12,21)(13,18)(14,19)(15,23)(17,22)(20,24);(9,12)(10,13)(11,17)(14,22)(15,20)(16,24)(18,23)(19,21);(9,13)(10,12)(11,18)(14,20)(15,22)(16,21)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=gi,b^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 | #72234480, #72234488, #72239419, … |
| maximal normal subgroup of | #72234480, #72234488, #72239419, … |
| maximal quotient of | #72261641, #72261672, #72261781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7589662);