Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500650 |
| 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 ◁ #2961 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
779 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,12)(4,18)(5,22)(6,21)(8,13)(9,23)(10,26)(14,30)(15,16)(17,24)(19,25)(20,27)(28,32)(29,31);(1,2)(3,13)(4,19)(5,23)(6,26)(7,11)(8,16)(9,22)(10,21)(12,28)(14,24)(15,32)(17,31)(18,25)(20,29)(27,30);(1,3)(2,8)(4,20)(5,24)(6,27)(7,15)(9,30)(10,14)(11,28)(12,18)(13,23)(16,22)(17,21)(19,32)(25,31)(26,29);(1,2)(3,14,15,29)(4,9)(5,25)(6,10)(7,11)(8,27,28,17)(12,31,16,30)(13,24,32,20)(18,23)(19,22)(21,26)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=gh,h^a=hi,c^b=cg,f^b=fhi,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=gh,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72213937, C2×#500650, #72249759, … |
| maximal normal subgroup of | #72213937, C2×#500650, #72249759, … |
| maximal quotient of | #72231517, #72231547, C2×#500650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407846);