Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6369150 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1703 conjugacy classes of subgroups; only the 494 normal subgroups are shown.
This group has 1703 conjugacy classes of subgroups — too many to draw. Only its 494 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,13)(5,7)(6,11)(8,14)(10,16)(12,15)(17,18)(19,23)(20,25)(21,24)(22,26)(27,29)(28,32)(30,31);(1,3)(2,6)(4,10)(5,12)(7,16)(8,15)(9,13)(11,14)(17,19,22,28)(18,23,26,32)(20,29,30,21)(24,25,27,31);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,16)(13,14)(18,24)(19,28)(21,29)(23,31)(25,32)(26,27);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(17,20)(18,25)(19,29)(21,28)(22,30)(23,27)(24,32)(26,31);(2,7)(6,16)(8,13)(9,15)(18,26)(19,28)(20,30)(24,27)Polycyclic presentation:
2^9|b^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72306584, #72306639, #72306690, … |
| maximal normal subgroup of | #72306584, #72306639, #72306690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276366);