Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(11,15)(14,16)(17,18,22,26)(19,27,29,24)(20,25,21,31)(23,30,32,28);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,25)(26,32)(27,31);(2,5)(6,8)(10,13)(14,16)(17,19)(18,24)(20,30)(21,28)(22,29)(23,25)(26,27)(31,32);(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,25,26,31)(19,28,29,30)(23,24,32,27)Polycyclic presentation:
2^9|a^2=i,c^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72285261, #72285313, #72313265, … |
| maximal normal subgroup of | #72285261, #72285313, #72313265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7051524);