Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417542 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,19,22,15)(12,16)(13,20,24,17)(14,18)(21,23);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,24)(17,23)(18,19);(2,6)(5,8)(9,12)(10,16)(11,20)(13,15)(14,23)(17,22)(18,21)(19,24);(9,13)(10,17)(11,21)(12,15)(14,24)(16,22)(18,20)(19,23)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,h^b=hi,e^c=ei,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72240906, #72266049, #72266050, … |
| maximal normal subgroup of | #72240906, #72266049, #72266050, … |
| maximal quotient of | #72240910, #72303387, #72303394 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358325);