Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144382 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10336 |
| commutator, Frattini, cosocle | C23×D4 | C23 |
| prime radical | #23897 | C2 |
| LCS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
957 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 957 conjugacy classes of subgroups — too many to draw. Only its 54 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)(3,11)(4,8)(5,17)(6,10)(7,13)(9,19)(12,26)(14,24)(15,25)(16,23)(18,21)(20,27)(22,30)(28,32)(29,31);(1,3)(2,7)(4,12)(5,18)(6,14)(8,20)(9,24)(10,21)(11,17)(13,19)(15,29)(16,28)(22,32)(23,31)(25,26)(27,30);(1,4,6,16)(2,8,10,23)(3,12,14,28)(5,15,19,30)(7,20,21,31)(9,25,17,22)(11,26,24,32)(13,29,18,27)Polycyclic presentation:
2^9|c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=gi,d^c=dg,e^d=ei,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72236521, #72237355, #72237377, … |
| maximal normal subgroup of | #72236521, #72237355, #72237377, … |
| maximal quotient of | #72236523, #72251078, #72337629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51579);