Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(14,18)(16,19)(22,25)(23,27);(5,7,9,10)(6,8,11,12)(13,14,17,18)(15,16,20,19)(23,27)(26,28);(13,15)(14,19)(16,18)(17,20)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,23,24,27)(22,26,25,28);(5,8)(6,10)(7,11)(9,12)(13,16)(14,20)(15,18)(17,19)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72326391, #72326394, #72326414 |
| maximal normal subgroup of | #72326391, #72326394, #72326414 |
| maximal quotient of | #72328501 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4773922);