Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #107114 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1041 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,19)(3,11)(4,15)(8,12,21,25)(10,24,23,32)(13,26)(14,28)(16,29)(18,27)(20,31,30,22);(1,2,4,10,5,9,16,23)(3,12,14,24,11,25,27,32)(6,8,15,22,17,21,29,31)(7,18,20,26,19,28,30,13);(3,13)(7,12)(11,26)(14,18)(19,25)(20,24)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72210458, #72213569, #72213643, … |
| maximal normal subgroup of | #72210458, #72213569, #72213643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 14312);