Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | C2×#2464 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,12,8,10)(13,14,18,16)(15,17,20,19)(23,27)(24,26);(5,7)(6,10)(8,9)(11,12)(13,15)(14,19)(16,17)(18,20)(21,22,25,28)(23,24,27,26);(7,8)(10,12)(14,16)(15,20)(21,23,25,27)(22,26,28,24);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,16,18,14)(15,17,20,19)(21,22)(23,26)(24,27)(25,28);(5,6)(7,10)(8,12)(9,11)(13,17)(14,20)(15,16)(18,19)(21,24,25,26)(22,27,28,23)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fh,d^2=f,e^2=fh,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2672248);