Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453994 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16255 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1849 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,11,5,21)(3,12,14,8,16,29,15,23)(4,9,20,26,19,28,18,27)(13,24,30,22,31,25,17,32);(1,3)(2,8)(4,17)(5,15)(6,16)(7,14)(9,24)(10,29)(11,23)(12,21)(13,20)(18,31)(19,30)(22,26)(25,28)(27,32);(1,4)(2,9)(3,13)(5,20)(6,19)(7,18)(8,22)(10,27)(11,28)(12,25)(14,17)(15,30)(16,31)(21,26)(23,32)(24,29);(1,3)(2,8)(4,17)(5,15)(6,16)(7,14)(9,25)(10,29)(11,23)(12,21)(13,20)(18,31)(19,30)(22,27)(24,28)(26,32)Polycyclic presentation:
2^9|a^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72216013, #72216154, … |
| maximal normal subgroup of | #72213514, #72216013, #72216154, … |
| maximal quotient of | #72237192, #72250614, C2×#454675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361871);