Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470225 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62773 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3399 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(3,9)(6,13)(10,15)(14,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(3,9)(6,13)(10,15)(14,16);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)Polycyclic presentation:
2^9|a^2=hi,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238024, #72251186, #72251191, … |
| maximal normal subgroup of | #72238024, #72251186, #72251191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403460);