Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
4223 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)(5,6,9,12)(7,11,8,10)(14,17)(15,19)(21,22,25,27)(23,24,28,26);(5,7,9,8)(6,10,12,11)(13,14,16,17)(15,18,19,20)(21,23,25,28)(22,26,27,24);(7,8)(10,11)(15,19)(18,20)(22,27)(24,26);(1,2)(3,4)(5,8,9,7)(6,11,12,10)(13,15,16,19)(14,18,17,20)(21,24,25,26)(22,23,27,28);(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,17)(18,19)(21,23)(22,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=gh,b^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232473, #72232491, #72232527, … |
| maximal normal subgroup of | #72232473, #72232491, #72232527, … |
| maximal quotient of | #72330591, #72330850, #72332902 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2977299);