Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3367 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)(5,6,8,9)(7,10,12,11)(13,14,16,17)(15,19,20,18)(22,24)(26,27);(5,6)(7,10)(8,9)(11,12)(14,17)(18,19)(21,22,25,24)(23,26,28,27);(6,9)(10,11)(13,14,16,17)(15,18,20,19)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,17)(18,20)(21,23)(22,27)(24,26)(25,28);(5,7)(6,10)(8,12)(9,11)(13,15,16,20)(14,18,17,19)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72327783, #72327801, #72327921 |
| maximal normal subgroup of | #72327783, #72327801, #72327921 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2668831);