Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499849 |
| 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 | |
1982 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)(7,16)(8,12)(9,13)(10,15)(11,19)(14,24)(17,33)(18,26)(20,30)(21,31)(22,27)(23,29)(25,36)(28,34)(32,35);(1,2)(3,4)(5,7,10,18)(6,11,15,27)(8,17,20,35)(9,22,24,19)(12,28,30,25)(13,16,14,26)(21,36,23,34)(29,32,31,33);(2,4)(5,8)(6,12)(7,17)(9,23)(10,20)(11,25)(13,31)(14,29)(15,30)(16,32)(18,35)(19,36)(21,24)(22,34)(26,33)(27,28);(6,13)(7,18)(8,20)(9,24)(11,26)(12,29)(14,15)(16,27)(25,32)(28,33)(30,31)(34,36)Polycyclic presentation:
2^9|b^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=eh,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72306393, #72306397, #72306398, … |
| maximal normal subgroup of | #72306393, #72306397, #72306398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407046);