Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2410 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,17,18)(15,19,20,16)(21,22,24,26)(23,27,28,25);(6,10)(7,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,25)(24,28)(26,27);(6,10)(8,11)(15,20)(16,19)(21,22,24,26)(23,25,28,27);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,23)(22,25)(24,28)(26,27);(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=fg,c^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72234680, #72234681, #72234745, … |
| maximal normal subgroup of | #72234680, #72234681, #72234745, … |
| maximal quotient of | #72328558, #72328688, #72328844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4506300);