Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5616 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)(6,9)(7,11)(14,17)(18,20)(21,22)(23,27)(24,28)(25,26);(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(22,26)(23,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,18,17,20)(21,23)(22,24)(25,28)(26,27);(21,24)(22,23)(25,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
| maximal subgroup of | #72326256, #72326259, #72326261, … |
| maximal normal subgroup of | #72326256, #72326259, #72326261, … |
| maximal quotient of | #72328392, #72328480, #72330274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 521852);