Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427246 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15368 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(4,17)(5,6)(8,22)(9,24)(11,12)(14,28)(15,20)(18,26)(19,21)(23,27)(29,32)(30,31);(1,2)(3,13)(4,18)(5,12)(6,11)(7,10)(8,15)(9,19)(14,29)(16,25)(17,24)(20,22)(21,26)(23,28)(27,30)(31,32);(1,3)(2,8)(4,14)(5,20)(6,15)(7,16)(9,23)(10,22)(11,25)(12,13)(17,28)(18,32)(19,30)(21,31)(24,29)(26,27);(1,4)(2,9,10,18)(3,14)(5,17)(6,21)(7,19)(8,23,22,32)(11,24,12,26)(13,27,25,29)(15,31)(16,30)(20,28)Polycyclic presentation:
2^9|d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216029, #72216037, … |
| maximal normal subgroup of | #72215958, #72216029, #72216037, … |
| maximal quotient of | C2×#427246, #72255621, #72265560 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334442);