Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24152 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
797 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)(3,6)(4,17)(5,16)(7,11)(8,13)(9,21)(10,23)(12,24)(14,19)(15,25)(18,32)(20,27)(22,30)(26,29)(28,31);(1,3,7,13,8,11,6,2)(4,18,20,28,22,14,26,9)(5,15,24,12,25,16,23,10)(17,31,29,32,30,21,27,19);(1,4)(2,9)(3,14)(5,19)(6,20)(7,26)(8,22)(10,27)(11,18)(12,29)(13,28)(15,30)(16,17)(21,24)(23,31)(25,32);(1,5,6,23,8,25,7,24)(2,10,11,16,13,12,3,15)(4,19,20,31,22,32,26,21)(9,27,18,17,28,29,14,30)Polycyclic presentation:
2^9|b^2=h,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,f^c=fh,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72214352, #72214675, #72214690, … |
| maximal normal subgroup of | #72214352, #72214675, #72214690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407798);