Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#447 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24208 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3450 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)(13,14,17,22,19,24,18,23)(15,21,27,20,25,28,16,26);(2,4)(6,10)(8,11)(13,15)(14,20)(16,17)(18,27)(19,25)(21,22)(23,28)(24,26);(5,6)(7,8)(9,10)(11,12)(13,16,19,27)(14,21,24,28)(15,17,25,18)(20,22,26,23);(5,7,9,12)(6,11,10,8)(13,17,19,18)(14,22,24,23)(15,16,25,27)(20,21,26,28);(5,8,9,11)(6,7,10,12)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh95 conjugacy classes — view the full character table.
| maximal subgroup of | #72308789, #72308978 |
| maximal normal subgroup of | #72308789, #72308978 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524615);