Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #492530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D82 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,19)(4,24)(5,27)(6,29)(7,20)(8,17)(9,16)(10,18)(11,21)(12,26)(13,28)(14,31)(15,32)(22,25)(23,30);(1,3,10,23)(2,11,18,28)(4,17,5,16)(6,24,32,27)(7,13,31,21)(8,12,9,22)(14,26,20,25)(15,30,29,19);(1,4)(2,12)(3,16)(5,10)(6,25)(7,30)(8,11)(9,28)(13,15)(14,27)(17,23)(18,22)(19,31)(20,24)(21,29)(26,32);(1,5)(2,13)(3,20)(4,10)(6,30)(7,25)(8,11)(9,28)(12,15)(14,23)(16,24)(17,27)(18,21)(19,32)(22,29)(26,31)Polycyclic presentation:
2^9|b^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gi,f^c=fh,h^c=hi,e^d=eg,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72214134, #72214187, #72214196, … |
| maximal normal subgroup of | #72214134, #72214187, #72214196, … |
| maximal quotient of | C2×#492532, #72216411, #72216416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 399728);