Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16697 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
885 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)(3,11)(4,8)(5,17,19,9)(6,10)(7,14)(12,20)(13,21,29,26)(15,22,18,24)(16,23)(25,28)(27,31,30,32);(1,3)(2,7,10,11)(4,12,16,28)(5,13)(6,14)(8,20)(9,21,17,26)(15,27,18,30)(19,29)(22,31)(23,25)(24,32);(1,4)(2,8)(3,12,14,28)(5,18,19,15)(6,16)(7,20,11,25)(9,24,17,22)(10,23)(13,27)(21,31)(26,32)(29,30);(1,5)(2,9)(3,13)(4,15)(6,19)(7,21)(8,22)(10,17)(11,26)(12,27)(14,29)(16,18)(20,31)(23,24)(25,32)(28,30)Polycyclic presentation:
2^9|a^2=eg,b^2=h,c^2=f,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237954, #72237956, #72237958, … |
| maximal normal subgroup of | #72237954, #72237956, #72237958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377932);