Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2078 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)(14,20)(15,22)(17,18)(23,24)(25,26)(27,28);(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,23,19,24,17,20)(15,21,22,27,25,16,26,28);(6,9)(10,11)(13,15,18,22,19,25,17,26)(14,21,23,27,24,16,20,28);(5,7,8,12)(6,10,9,11)(13,16)(14,22)(15,20)(17,27)(18,28)(19,21)(23,25)(24,26);(15,25)(16,21)(22,26)(27,28)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249920, #72256992, #72257004, … |
| maximal normal subgroup of | #72249920, #72256992, #72257004, … |
| maximal quotient of | #72264315 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370993);