Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2404 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,8,9)(7,11,12,10)(14,17)(15,19)(21,22,25,28)(23,24,27,26);(6,9)(7,12)(13,14,16,17)(15,20,19,18)(21,23,25,27)(22,26,28,24);(5,7,8,12)(6,10,9,11)(13,14,16,17)(15,18,19,20)(21,24,25,26)(22,27,28,23);(1,2,4,3)(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,26)(27,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=f,d^2=fghi,e^2=fghi,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232593, #72232623, #72232652, … |
| maximal normal subgroup of | #72232593, #72232623, #72232652, … |
| maximal quotient of | #72293980, C2×#572764, #72328462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 479960);