Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
3713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,11)(10,12)(13,14,17,16)(15,18,19,20)(22,25)(26,27);(1,2,3,4)(6,7)(8,12)(14,16)(18,20)(21,22)(23,27)(24,25)(26,28);(1,2)(3,4)(6,7)(10,11)(15,19)(18,20)(22,25)(26,27);(5,7,9,6)(8,11,12,10)(13,15,17,19)(14,18,16,20)(21,22)(23,26)(24,25)(27,28);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,20,19,18)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=fi,b^2=ghi,d^2=fgi,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72278272, #72278611, C2×#1892920, … |
| maximal normal subgroup of | #72278272, #72278611, C2×#1892920, … |
| maximal quotient of | #72278726, #72278770, #72278959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1800116);