Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2543 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)(5,6)(7,10)(8,9)(11,12)(13,14,17,23,18,22,19,20)(15,24,26,27,16,21,25,28);(2,4)(14,20)(15,25)(16,26)(17,19)(21,24)(22,23);(6,9)(10,12)(13,15,19,25,18,16,17,26)(14,21,20,27,22,24,23,28);(13,16,19,26,18,15,17,25)(14,21,20,27,22,24,23,28);(5,7,8,11)(6,10,9,12)(14,22)(15,16)(20,23)(25,26)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72280255, #72282866, #72282925, … |
| maximal normal subgroup of | #72280255, #72282866, #72282925, … |
| maximal quotient of | #72311638, #72312039 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374809);