Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244499 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1154 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 148 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:
(1,2,7,3)(4,15,19,26)(5,9,20,11)(6,21,24,27)(8,12,23,14)(10,22,13,16)(17,30,32,31)(18,28,29,25);(2,9)(3,11)(5,20)(6,24)(10,15)(12,21)(13,26)(14,27)(16,22)(18,29)(25,30)(28,31);(1,3,7,2)(4,13,19,10)(5,21,20,27)(6,9,24,11)(8,14,23,12)(15,29,26,18)(16,31,22,30)(17,28,32,25);(1,4)(2,10)(3,13)(5,22)(6,18)(7,19)(8,17)(9,15)(11,26)(12,25)(14,28)(16,20)(21,30)(23,32)(24,29)(27,31)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,g^c=gh,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72267849 |
| maximal normal subgroup of | #72267849 |
| maximal quotient of | #72267853 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151698);