Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396104 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14730 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,15,8)(4,17,18,32)(5,21)(6,11)(7,10)(9,20,25,19)(13,22,16,24)(14,28)(23,29)(26,27)(30,31);(1,3)(2,8)(4,14)(5,13)(6,16)(7,15)(9,23)(10,12)(11,22)(17,31)(18,30)(19,27)(20,29)(21,24)(25,26)(28,32);(3,13)(5,6)(8,22)(11,21)(12,24)(14,27)(15,16)(17,32)(19,20)(23,31)(26,28)(29,30);(1,4)(2,9,10,25)(3,14)(5,19)(6,20)(7,18)(8,23,12,26)(11,17,21,32)(13,27)(15,30)(16,29)(22,31,24,28)Polycyclic presentation:
2^9|a^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216047, #72216130, #72216166, … |
| maximal normal subgroup of | #72216047, #72216130, #72216166, … |
| maximal quotient of | C2×#396104, #72255870, #72265718 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303300);