Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393907 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3883 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 3883 conjugacy classes of subgroups — too many to draw. Only its 296 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:
(2,8)(3,12)(5,6)(10,25)(11,20)(13,21)(14,29)(15,16)(17,18)(22,26)(27,30)(28,31);(1,2)(3,13)(4,10)(5,20)(6,8)(7,11)(9,15)(12,24)(14,28)(16,21)(17,22)(18,25)(19,26)(23,29)(27,32)(30,31);(1,3)(2,9)(4,14)(5,12)(6,15)(7,16)(8,21)(10,23)(11,24)(13,20)(17,29)(18,27)(19,30)(22,31)(25,28)(26,32);(1,4)(2,10)(3,14)(5,18)(6,17)(7,19)(8,22)(9,23)(11,26)(12,27)(13,28)(15,29)(16,30)(20,25)(21,31)(24,32)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72241161, #72241179, #72241303, … |
| maximal normal subgroup of | #72241161, #72241179, #72241303, … |
| maximal quotient of | C2×#393907 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301103);