Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator, socle | C23 | C22×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1484 conjugacy classes of subgroups; only the 594 normal subgroups are shown.
This group has 1484 conjugacy classes of subgroups — too many to draw. Only its 594 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,8,12,9,11,15,18)(7,10,14,17,13,16,19,20)(21,22)(23,26)(24,28)(25,27)(29,32)(30,33)(31,34)(35,36);(2,4)(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,24,30,25,29,31,35)(22,26,28,33,27,32,34,36);(6,11)(7,13)(12,18)(14,19)(23,29)(26,32)(30,35)(33,36);(7,13)(10,16)(14,19)(17,20)(22,27)(26,32)(28,34)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,c^b=chi,d^b=dh104 conjugacy classes — view the full character table.
| derived subgroup of | #72238321, #72238323, #72239350, … |
| Fitting subgroup of | #72238321 |
| maximal subgroup of | #72238289, #72238321, #72239297, … |
| maximal normal subgroup of | #72238289, #72238321, #72239297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88053);