Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×#509 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(6,11)(7,15)(8,19)(9,22)(12,28)(13,27)(14,31)(16,29)(17,34)(18,25)(20,24)(23,35)(26,30)(32,33);(5,6)(7,16)(8,20)(9,24)(10,13)(11,19)(12,17)(14,22)(15,33)(18,28)(21,32)(23,31)(25,26)(27,35)(29,34)(30,36);(2,4);(5,7,10,18)(6,12,13,29)(8,21,23,36)(9,25,22,15)(11,26,27,33)(14,32,24,30)(16,31,28,20)(17,35,34,19);(6,13)(11,27)(12,29)(14,24)(16,28)(20,31)(26,33)(30,32)Polycyclic presentation:
2^9|d^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286631, D4×#3414, #72320189, … |
| maximal normal subgroup of | #72286631, D4×#3414, #72320189, … |
| maximal quotient of | #72287577, #72309456, #72309719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530819);