Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499746 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23848 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2931 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1312 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)(3,15)(4,8)(5,14)(6,12)(7,13)(9,17)(10,16)(11,19)(18,30)(20,24)(21,26)(22,25)(23,29)(27,32)(28,31);(2,8)(4,19)(5,6)(9,27)(10,15)(11,14)(12,26)(13,25)(16,20)(17,23)(21,22)(24,29)(28,30)(31,32);(1,3)(2,9)(4,16)(5,23)(6,17)(7,18)(8,24)(10,26)(11,28)(12,30)(13,29)(14,15)(19,20)(21,32)(22,31)(25,27);(1,3)(2,10)(4,20)(5,17)(6,23)(7,18)(8,15)(9,26)(11,29)(12,27)(13,28)(14,24)(16,19)(21,31)(22,32)(25,30)Polycyclic presentation:
2^9|e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=eh,f^b=fi,h^b=hi,f^c=fi,e^d=eh,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213945, #72231612, #72231620, … |
| maximal normal subgroup of | #72213945, #72231612, #72231620, … |
| maximal quotient of | #72224952, #72237256, #72250557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406942);