Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493397 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62842 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3493 conjugacy classes of subgroups; only the 328 normal subgroups are shown.
This group has 3493 conjugacy classes of subgroups — too many to draw. Only its 328 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)(3,11,14,24)(4,8)(5,17,19,9)(6,10)(7,13)(12,25,26,32)(15,22)(16,29,28,23)(18,21)(20,27)(30,31);(1,3)(2,7)(4,12)(5,18)(6,14)(8,20)(9,24)(10,21)(11,17)(13,19)(15,26)(16,30)(22,31)(23,32)(25,29)(27,28);(4,15)(5,19)(8,22)(9,17)(11,24)(12,26)(13,18)(20,31);(1,4,6,15)(2,8,10,22)(3,12,14,26)(5,16,19,28)(7,20,21,31)(9,23,17,29)(11,25,24,32)(13,27,18,30)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fi,g^b=ghi,d^c=di,e^c=ei,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72210453, #72210522, #72210534, … |
| maximal normal subgroup of | #72210453, #72210522, #72210534, … |
| maximal quotient of | C2×#493398, #72258026, #72266291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400594);