Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479880 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17301 |
| commutator, Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
635 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 635 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 32 points:
(1,2)(3,11)(4,8,17,24)(5,18,20,9)(6,10)(7,14)(12,21,15,25)(13,26,19,22)(16,23)(27,32)(28,31)(29,30);(1,3)(2,7,10,11)(4,15,17,12)(5,19)(6,14)(8,21)(9,22,18,26)(13,20)(16,28,29,27)(23,32)(24,25)(30,31);(1,4)(2,8,10,24)(3,12)(5,16)(6,17)(7,21,11,25)(9,23,18,30)(13,28)(14,15)(19,27)(20,29)(22,32,26,31);(1,5)(2,9)(3,13,14,19)(4,16)(6,20)(7,22,11,26)(8,23)(10,18)(12,27,15,28)(17,29)(21,31,25,32)(24,30)Polycyclic presentation:
2^9|a^2=fg,b^2=eh,c^2=eg,d^2=e,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72318288, #72318318, #72318394, … |
| maximal normal subgroup of | #72318288, #72318318, #72318394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387076);