Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499885 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60388 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1860 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)(3,9)(4,17)(5,8)(6,22)(7,12)(10,19)(11,21)(13,29)(14,26)(15,30)(16,28)(18,32)(20,25)(23,27)(24,31);(2,8)(3,13)(5,21)(6,19)(9,26)(10,17)(11,12)(14,31)(15,18)(16,23)(22,25)(24,27)(28,32)(29,30);(1,3,7,16)(2,9,12,28)(4,18,20,14)(5,15,21,31)(6,23,19,13)(8,24,11,30)(10,29,22,27)(17,26,25,32);(1,4,7,20)(2,10,12,22)(3,14,16,18)(5,6,21,19)(8,17,11,25)(9,27,28,29)(13,31,23,15)(24,32,30,26)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72236917, #72237519, #72237525, … |
| maximal normal subgroup of | #72237519, #72237525, #72237532, … |
| maximal quotient of | #72311964, #72312095, #72312679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407083);