Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #44933 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(11,15)(14,16)(17,18,22,25)(19,21,24,31)(20,23,30,32)(26,29,27,28);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,25,22,24,31,18)(20,27,29,23,30,26,28,32);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,27)(25,32)(28,31);(9,11,12,15)(10,14,13,16)(17,21,22,31)(18,24,25,19)(20,28,30,29)(23,26,32,27)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280975, #72281059, #72281126, … |
| maximal normal subgroup of | #72280975, #72281059, #72281126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7064186);