Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3545 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,18,28)(8,12)(9,21,23,13)(10,14)(11,17)(16,31,32,30)(19,26)(20,29,34,27)(22,25)(24,33)(35,36);(5,7,10,18)(6,11,14,25)(8,16,19,32)(9,22,23,17)(12,24,26,36)(13,28,21,15)(20,35,34,33)(27,30,29,31);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,29)(14,26)(15,30)(17,33)(18,32)(21,27)(22,35)(23,34)(25,36)(28,31);(8,19)(12,26)(16,32)(20,34)(24,36)(27,29)(30,31)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fhi,g^b=ghi,d^c=di,e^c=eh,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72234427, #72260670, #72279632, … |
| maximal normal subgroup of | #72234427, #72260670, #72279632, … |
| maximal quotient of | #72234428 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400828);