Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17101 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,16,8,15,24,17,25)(4,18,20,27,22,28,21,9)(14,30,19,26,31,23,32,29);(1,3,6,16,7,15,5,17)(2,8,10,24,12,25,11,13)(4,19,20,31,22,32,21,14)(9,26,18,23,27,29,28,30);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,23)(10,28)(11,18)(12,27)(13,29)(15,31)(16,32)(17,19)(24,26)(25,30);(2,10)(3,15)(4,20)(5,6)(8,24)(9,27)(11,12)(13,25)(14,19)(21,22)(26,29)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72339679, #72339680, #72339684 |
| maximal normal subgroup of | #72339679, #72339680, #72339684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384195);