Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500646 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24196 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
804 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)(3,12)(4,17)(5,22)(6,11)(7,13)(8,18)(9,28)(10,20)(14,19)(15,24)(16,30)(21,29)(23,27)(25,32)(26,31);(3,13)(4,18)(5,20)(7,24)(8,17)(9,29)(10,22)(12,25)(14,30)(15,32)(16,31)(19,23)(21,28)(26,27);(1,3)(2,7)(4,19)(5,23)(6,15)(8,27)(9,30)(10,14)(11,25)(12,17)(13,18)(16,20)(21,31)(22,26)(24,29)(28,32);(1,2)(3,14,15,26)(4,8)(5,10)(6,11)(7,23,25,16)(9,21)(12,31,24,19)(13,30,32,27)(17,18)(20,22)(28,29)Polycyclic presentation:
2^9|d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=gh,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, C2×#500646 |
| maximal normal subgroup of | #72213945, C2×#500646 |
| maximal quotient of | #72226810, #72239911, #72250663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407842);