Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500473 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24128 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(7,24)(11,27)(12,25)(26,28);(1,2)(3,11)(4,16)(5,22)(6,8)(7,14)(9,20)(10,19)(12,17)(13,27)(15,28)(18,30)(21,31)(23,26)(24,29)(25,32);(1,3)(2,7)(4,17)(5,23)(6,13)(8,24)(9,26)(10,12)(11,16)(14,20)(15,19)(18,28)(21,32)(22,25)(27,31)(29,30);(1,2)(3,12,13,25)(4,18)(5,10)(6,8)(7,23,24,15)(9,21)(11,17,27,32)(14,26,29,28)(16,30)(19,22)(20,31)Polycyclic presentation:
2^9|d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=gh,h^c=hi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, C2×#500473, #72249766, … |
| maximal normal subgroup of | #72213937, C2×#500473, #72249766, … |
| maximal quotient of | #72231488, #72231535, C2×#500473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407669);