Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130780 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8375 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #61079 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
833 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 833 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,13,6,25,3,15,9,14,26,31,8,12,18,30)(4,19,21,11,23,28,24,32,17,29,5,22,20,10,16,27);(1,3)(2,10)(4,16)(5,23)(6,7)(8,26)(9,18)(11,13)(12,19)(14,28)(15,27)(17,24)(20,21)(22,31)(25,29)(30,32);(1,4)(2,10)(3,16)(5,7)(6,23)(8,20)(9,17)(11,15)(12,29)(13,27)(14,28)(18,24)(19,25)(21,26)(22,30)(31,32)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fgi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231620, #72231700, #72231704, … |
| maximal normal subgroup of | #72231620, #72231700, #72231704, … |
| maximal quotient of | C2×#130782, #72250844, #72280195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37978);