Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401631 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14820 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,13,6,11)(3,14,17,8,19,26,18,27)(4,20,21,10,22,29,23,9)(15,24,16,28,32,30,31,25);(1,3,5,17,7,19,6,18)(2,8,11,14,13,27,12,26)(4,16,21,32,22,31,23,15)(9,28,29,24,10,25,20,30);(1,4)(2,9)(3,15)(5,23)(6,21)(7,22)(8,24)(10,13)(11,20)(12,29)(14,28)(16,18)(17,31)(19,32)(25,26)(27,30);(1,4)(2,10)(3,16)(5,21)(6,23)(7,22)(8,25)(9,13)(11,20)(12,29)(14,30)(15,18)(17,32)(19,31)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216037, #72216154, #72216217, … |
| maximal normal subgroup of | #72216037, #72216154, #72216217, … |
| maximal quotient of | C2×#401631, #72252173, #72255626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308827);