Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23 | C43 |
| Frattini, prime radical, cosocle | C26 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C2 ▷ C1 | |
2718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,14,18,21)(12,15)(13,16)(17,23,24,20)(19,22);(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,23)(19,24);(2,4)(5,6,7,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,b^a=bd,c^a=ce,d^a=di,g^a=gi,d^b=di,e^b=ei,f^b=fi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72265635, #72265641, #72265648, … |
| maximal normal subgroup of | #72265635, #72265641, #72265648, … |
| maximal quotient of | #72266031, #72268608, #72268625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30689);