Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #469730 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16661 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1090 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)(3,7)(4,8)(5,9,18,25)(6,10)(11,26)(12,28,30,20)(13,21)(14,19)(15,22,17,24)(16,23)(27,31,29,32);(1,3)(2,7)(4,14)(5,12)(6,13)(8,19)(9,20)(10,21)(11,16)(15,29)(17,27)(18,30)(22,31)(23,26)(24,32)(25,28);(1,4)(2,8)(3,11,13,14)(5,17)(6,16)(7,19,21,26)(9,24)(10,23)(12,29,30,27)(15,18)(20,32,28,31)(22,25);(1,5)(2,9,10,25)(3,12)(4,15,16,17)(6,18)(7,20,21,28)(8,22)(11,27,14,29)(13,30)(19,31)(23,24)(26,32)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=g,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,d^c=de,g^c=gi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237762, #72237765, #72237767, … |
| maximal normal subgroup of | #72237762, #72237765, #72237767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376926);