Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623665 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,8)(3,13)(4,16)(5,6)(9,26)(10,27)(11,25)(12,30)(14,31)(15,21)(17,22)(19,20)(23,28)(24,29);(1,2)(3,9)(4,17)(5,12)(6,22)(7,11)(8,16)(10,14)(13,28)(15,27)(18,23)(19,25)(20,30)(21,24)(26,31)(29,32);(1,3)(2,9)(4,14)(5,21)(6,13)(7,15)(8,23)(10,17)(11,27)(12,24)(16,18)(19,32)(20,31)(22,28)(25,29)(26,30);(1,3)(2,10)(4,18)(5,13)(6,21)(7,15)(8,24)(9,17)(11,26)(12,23)(14,16)(19,31)(20,32)(22,29)(25,28)(27,30);(2,11)(8,25)(9,27)(10,26)(12,22)(17,30)(23,29)(24,28)Polycyclic presentation:
2^9|f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,g^c=gi,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72214134, #72214153, … |
| maximal normal subgroup of | #72214039, #72214134, #72214153, … |
| maximal quotient of | #72216402, #72226906, D8×#509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530861);