Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,18,19,33)(9,13,21,27)(10,14)(12,20)(15,28,29,32)(16,23)(17,24,31,26)(22,30)(25,34)(35,36);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,22)(13,26)(14,23)(18,32)(19,29)(20,30)(21,31)(24,27)(25,35)(28,33)(34,36);(5,8,10,19)(6,12,14,25)(7,15,16,29)(9,20,21,34)(11,22,23,35)(13,18,27,33)(17,30,31,36)(24,28,26,32);(7,16)(11,23)(15,29)(17,31)(22,35)(24,26)(28,32)(30,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fhi,e^c=eh,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279139, #72279160, #72279206, … |
| maximal normal subgroup of | #72279139, #72279160, #72279206, … |
| maximal quotient of | #72317686, #72317796, #72317804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401772);