Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#465 |
| 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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14,17,21)(15,20,23,18)(16,24,25,27)(19,26,28,22);(1,2)(3,4)(6,10)(11,12)(13,15,17,23)(14,18,21,20)(16,22,25,26)(19,24,28,27);(2,4)(5,7,9,8)(6,11,10,12)(13,16,15,22,17,25,23,26)(14,19,18,24,21,28,20,27);(1,2)(3,4)(5,7,9,8)(6,11,10,12);(2,4)(5,8,9,7)(6,12,10,11)(13,15,17,23)(14,20,21,18)(16,22,25,26)(19,27,28,24)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=eh110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492784);