Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(9,12)(13,14,17,19)(15,18,16,20)(21,22,24,26)(23,25,28,27);(5,6,8,10)(7,9,11,12)(13,15,17,16)(14,18,19,20)(21,22)(23,25)(24,26)(27,28);(5,6)(7,9)(8,10)(11,12)(13,16,17,15)(14,18,19,20)(21,23,24,28)(22,25,26,27);(1,2)(3,4)(13,14)(15,20)(16,18)(17,19)(21,22)(23,27)(24,26)(25,28);(5,7)(6,9)(8,11)(10,12)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72332336, #72332344, #72332358, … |
| maximal normal subgroup of | #72332336, #72332344, #72332358, … |
| maximal quotient of | #72332519 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4360478);