Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(10,14)(11,15)(17,18)(19,20)(21,22)(23,24)(25,26,29,27)(28,30,32,31);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,21,22)(19,20,23,24)(26,27)(30,31);(3,7)(6,8)(10,14)(12,16)(18,22)(20,24)(25,26,29,27)(28,31,32,30);(1,2)(3,8)(4,5)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27,29,26)(28,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,20)(18,23)(19,22)(21,24)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=fgi,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232610, #72232624, #72232629, … |
| maximal normal subgroup of | #72232610, #72232624, #72232629, … |
| maximal quotient of | #72328438, #72328763, #72329888 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3454331);