Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128061 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8101 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1375 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1375 conjugacy classes of subgroups — too many to draw. Only its 140 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:
(1,2,5,10,6,9,19,24)(3,7,12,21,13,22,17,30)(4,14,16,25,15,8,18,23)(11,20,27,31,26,29,28,32);(1,3)(2,7,9,22)(4,11)(5,17,19,12)(6,13)(8,20,14,29)(10,21)(15,26)(16,28,18,27)(23,31)(24,30)(25,32);(1,4)(2,8)(3,11)(5,18)(6,15)(7,20)(9,14)(10,25)(12,28)(13,26)(16,19)(17,27)(21,32)(22,29)(23,24)(30,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fh,g^a=gi,h^a=hi,d^b=dh,e^b=eg,g^b=gi,d^c=di,e^c=ei,e^d=ei,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72210444, #72210553, #72210561, … |
| maximal normal subgroup of | #72210444, #72210553, #72210561, … |
| maximal quotient of | #72234506, #72235163, #72235167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35259);