Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499803 |
| 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 |
| prime radical | C2×#3026 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2051 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)(11,18)(16,31)(17,24)(19,27)(20,28)(21,36)(22,29)(23,35)(25,34)(26,33)(30,32);(1,2)(3,4)(5,7,10,17)(6,11,14,25)(8,16,20,32)(9,18,22,34)(12,26,28,23)(13,15,29,24)(19,33,21,35)(27,31,36,30);(2,4)(5,8,10,20)(6,12,14,28)(7,16,17,32)(9,21,22,19)(11,23,25,26)(13,27,29,36)(15,30,24,31)(18,35,34,33);(6,13)(7,17)(11,24)(12,27)(14,29)(15,25)(16,32)(18,34)(23,31)(26,30)(28,36)(33,35)Polycyclic presentation:
2^9|b^2=i,c^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fh,e^c=eh,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72303827, #72303830, #72303838, … |
| maximal normal subgroup of | #72303827, #72303830, #72303838, … |
| maximal quotient of | #72303880, #72303983 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407002);