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 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2025 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)(7,17)(8,13)(9,24)(10,15)(11,16)(12,19)(14,20)(18,29)(21,28)(22,30)(23,31)(25,27)(26,34)(32,36)(33,35);(1,2)(3,4)(5,7,11,20)(6,12,16,24)(8,18,23,36)(9,10,19,26)(13,28,31,27)(14,15,17,34)(21,35,25,22)(29,33,32,30);(2,4)(5,8,11,23)(6,13,16,31)(7,18,20,36)(9,25,19,21)(10,22,26,35)(12,27,24,28)(14,32,17,29)(15,33,34,30);(5,9,11,19)(6,14,16,17)(7,10,20,26)(8,21,23,25)(12,15,24,34)(13,29,31,32)(18,35,36,22)(27,30,28,33)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^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 | #72279120, #72279160, #72279238, … |
| maximal normal subgroup of | #72279120, #72279160, #72279238, … |
| maximal quotient of | #72303931 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406998);