Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #440904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,21,7,9,6,11)(3,12,13,24,15,8,16,22)(4,17,20,25,18,32,19,10)(14,28,29,23,30,31,27,26);(1,3,5,16,7,15,6,13)(2,8,21,24,9,12,11,22)(4,14,19,29,18,30,20,27)(10,26,32,28,25,23,17,31);(2,9)(3,13)(5,6)(8,22)(10,25)(12,24)(14,27)(15,16)(19,20)(23,28)(26,31)(29,30);(1,4)(2,10,9,25)(3,14)(5,19)(6,20)(7,18)(8,23,12,26)(11,17,21,32)(13,27)(15,30)(16,29)(22,28,24,31)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72216029, #72216077, #72216217, … |
| maximal normal subgroup of | #72216029, #72216077, #72216217, … |
| maximal quotient of | C2×#440904, #72252168, #72255809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348100);