Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1093 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)(3,13)(4,18)(5,20,21,10)(6,11)(7,12)(8,16,25,32)(9,19,27,22)(14,28)(15,24)(17,29)(23,30,26,31);(1,3,7,17)(2,8,12,25)(4,15,6,14)(5,16,21,32)(9,24,27,28)(10,13,20,29)(11,23,18,26)(19,31,22,30);(3,14)(4,6)(5,21)(8,23)(9,27)(11,18)(13,28)(15,17)(16,31)(24,29)(25,26)(30,32);(1,4,7,6)(2,9,12,27)(3,15,17,14)(5,19,21,22)(8,24,25,28)(10,18,20,11)(13,26,29,23)(16,31,32,30)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^b=fh,g^b=gi,d^c=di,e^c=ehi,g^c=gi,f^d=fh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72237519, #72237523, #72237549, … |
| maximal normal subgroup of | #72237519, #72237523, #72237549, … |
| maximal quotient of | #72311939, #72312115, #72312368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408397);