Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468845 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16632 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,5,12)(3,13,16,23,17,24,15,8)(4,18,20,9,21,27,19,26)(14,22,30,28,31,25,29,32);(1,3)(2,8,10,23)(4,14)(5,15)(6,16)(7,17)(9,25,26,22)(11,13,12,24)(18,28,27,32)(19,29)(20,30)(21,31);(1,4)(2,9,10,26)(3,14)(5,19)(6,20)(7,21)(8,22,23,25)(11,27,12,18)(13,28,24,32)(15,29)(16,30)(17,31);(2,10)(4,19)(5,6)(8,23)(9,18)(14,29)(15,16)(20,21)(22,32)(25,28)(26,27)(30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72216037, #72216130, #72216203, … |
| maximal normal subgroup of | #72216037, #72216130, #72216203, … |
| maximal quotient of | C2×#468845, #72255623, #72255648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376041);