Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433042 |
| 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 |
| prime radical | #29736 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1664 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(19,25)(20,26)(22,28)(24,29)(27,31)(30,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,23)(19,22,27,30)(20,24,26,29)(25,28,31,32);(11,15)(13,16)(17,19,26,31,21,27,20,25)(18,22,29,32,23,30,24,28);(10,14)(13,16)(18,23)(22,30)(24,29)(28,32);(1,3,4,7)(2,6,5,8)(19,27)(22,30)(25,31)(28,32)Polycyclic presentation:
2^9|b^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257627, #72264175, #72264267, … |
| maximal normal subgroup of | #72257627, #72264175, #72264267, … |
| maximal quotient of | #72265104 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341225);