Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499795 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,32)(18,31)(19,28)(20,27)(21,26)(22,29)(23,35)(24,36)(25,34)(30,33);(1,2)(3,4)(5,7,10,18)(6,11,14,24)(8,16,19,33)(9,17,22,36)(12,25,28,23)(13,15,29,31)(20,34,21,35)(26,30,27,32);(2,4)(5,8)(6,12)(7,16)(9,21)(10,19)(11,23)(13,27)(14,28)(15,30)(17,35)(18,33)(20,22)(24,25)(26,29)(31,32)(34,36);(6,13)(8,19)(11,15)(12,26)(14,29)(16,33)(20,21)(23,32)(24,31)(25,30)(27,28)(34,35)Polycyclic presentation:
2^9|b^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,d^c=di,e^c=eh,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72279131, #72279224, #72279238, … |
| maximal normal subgroup of | #72279131, #72279224, #72279238, … |
| maximal quotient of | #72317668, #72317723, #72317727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406992);