Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 | C2×#459 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(15,23)(16,17)(19,26)(21,22)(24,25)(27,28);(1,2,3,4)(5,6,10,8)(7,12,9,11)(13,14,18,20)(15,19,24,27)(16,21,17,22)(23,26,25,28);(5,7,10,9)(6,11,8,12)(13,15,18,24)(14,19,20,27)(16,23,17,25)(21,26,22,28);(5,8,10,6)(7,11,9,12)(14,20)(19,27)(21,22)(26,28);(5,9,10,7)(6,12,8,11)(13,16,18,17)(14,21,20,22)(15,23,24,25)(19,26,27,28)Polycyclic presentation:
2^9|b^2=fhi,c^2=hi,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257316, #72257697, #72263848, … |
| maximal normal subgroup of | #72257316, #72257697, #72263848, … |
| maximal quotient of | #72265033 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6532864);