Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144640 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10391 |
| commutator, Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 46 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:
(1,2,5,9)(3,11,13,7)(4,8,16,22)(6,10,18,25)(12,28,30,19)(14,27,31,20)(15,26,32,21)(17,23,29,24);(1,3,5,13)(2,7,9,11)(4,15,16,32)(6,14,18,31)(8,21,22,26)(10,20,25,27)(12,29,30,17)(19,24,28,23);(1,4)(2,8,10,23)(3,12,14,15)(5,16)(6,17)(7,19)(9,24,25,22)(11,26)(13,30,31,32)(18,29)(20,21)(27,28)Polycyclic presentation:
2^9|a^2=d,b^2=d,c^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,h^b=hi,d^c=dg,e^c=ei,f^c=fi,e^d=ei29 conjugacy classes — view the full character table.
| maximal subgroup of | #72235142, #72285653, #72285683, … |
| maximal normal subgroup of | #72235142, #72285653, #72285683, … |
| maximal quotient of | #72235171, #72235243, #72235247 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51836);