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) = C26×C4 |
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 | |
3370 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)(15,19)(18,20)(21,22)(23,28)(24,26)(25,27);(2,4)(5,6,8,9)(7,10,11,12)(13,14,17,16)(15,18,19,20)(21,23,25,24)(22,26,27,28);(1,2)(3,4)(6,9)(10,12)(22,27)(26,28);(5,6)(7,12)(8,9)(10,11)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,28)(25,27);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,20,19,18)(21,24,25,23)(22,26,27,28)Polycyclic presentation:
2^9|b^2=gi,e^2=gi,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 | #72332348, #72332352, #72332378 |
| maximal normal subgroup of | #72332348, #72332352, #72332378 |
| maximal quotient of | #72332939 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2696873);