Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438955 |
| 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 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2430 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)(7,11)(8,9)(10,12)(13,14,18,20,19,24,17,23)(15,21,25,27,16,22,26,28);(2,4)(6,9)(10,11)(14,20)(15,25)(16,26)(17,18)(21,22)(23,24);(5,7,8,12)(6,10,9,11)(13,15,18,25,19,16,17,26)(14,21,20,27,24,22,23,28);(13,16,18,26,19,15,17,25)(14,22,20,28,24,21,23,27);(15,16)(21,22)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fgi,c^2=gh,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,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 | #72279952, #72280126, #72280255, … |
| maximal normal subgroup of | #72279952, #72280126, #72280255, … |
| maximal quotient of | #72313047, #72313054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373399);