Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15323 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1252 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)(3,13,15,8)(4,9)(5,12)(6,10)(7,11)(14,22,30,29)(16,23,17,24)(18,25,31,32)(19,28)(20,27)(21,26);(1,3)(2,8)(4,18)(5,17)(6,16)(7,15)(9,25)(10,24)(11,13)(12,23)(14,20)(19,31)(21,30)(22,27)(26,29)(28,32);(1,4)(2,9)(3,14,15,30)(5,20)(6,21)(7,19)(8,22,13,29)(10,26)(11,28)(12,27)(16,18,17,31)(23,32,24,25);(2,10)(5,6)(8,23)(9,26)(11,12)(13,24)(16,17)(18,31)(20,21)(22,32)(25,29)(27,28)Polycyclic presentation:
2^9|a^2=f,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72216029, #72216077, #72216166, … |
| maximal normal subgroup of | #72216029, #72216077, #72216166, … |
| maximal quotient of | C2×#424613, #72255858, #72265119 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331809);