Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
4098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14,17,19)(15,16,20,18)(21,22)(23,28)(24,26)(25,27);(2,4)(5,6)(7,12)(8,9)(10,11)(13,15,17,20)(14,18,19,16)(21,23)(22,25)(24,28)(26,27);(1,2,3,4)(6,9)(10,12)(14,19)(15,20)(22,24)(23,27);(5,6)(7,12)(8,9)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24,26,22)(23,25,27,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,20)(15,19)(17,18)(21,25)(22,27)(23,24)(26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fh,d^2=fgi,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 | #72260786, #72260787, #72260826, … |
| maximal normal subgroup of | #72260786, #72260787, #72260826, … |
| maximal quotient of | #72323676 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3068378);