Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3922 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)(8,9)(10,12)(13,14,18,19)(15,16,17,20)(21,22,25,24)(23,27,28,26);(6,8)(10,11)(13,15,18,17)(14,16,19,20)(21,23)(22,26)(24,27)(25,28);(7,12)(10,11)(14,19)(16,20)(21,23,25,28)(22,27,24,26);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,20)(14,15,19,17)(21,23)(22,27)(24,26)(25,28);(5,8,9,6)(7,11,12,10)(13,17,18,15)(14,20,19,16)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300484, #72300498, #72300545, … |
| maximal normal subgroup of | #72300484, #72300498, #72300545, … |
| maximal quotient of | #72330074, #72332912 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4894196);