Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511849 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32899 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2142 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)(3,9)(4,11)(5,10)(6,21)(7,12)(8,14)(13,29)(15,24)(16,25)(17,27)(18,22)(19,32)(20,26)(23,31)(28,30);(1,3)(2,8)(4,13)(5,14)(6,16)(7,15)(9,10)(11,28)(12,22)(17,30)(18,20)(19,31)(21,23)(24,26)(25,32)(27,29);(1,3)(2,9)(4,16)(5,18)(6,13)(7,15)(8,10)(11,25)(12,24)(14,20)(17,30)(19,31)(21,29)(22,26)(23,27)(28,32);(1,4,7,6)(3,13,15,16)(5,19,20,17)(9,24)(10,26)(14,31,18,30)(25,29)(27,32)Polycyclic presentation:
2^9|d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,f^b=fhi,g^b=gi,d^c=di,e^c=eh,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72260481, #72260545 |
| maximal normal subgroup of | #72260481, #72260545 |
| maximal quotient of | #72261123, #72261277, #72262728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419045);