Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,11,8,7,10,12)(14,18)(15,22)(16,19)(20,27)(21,25)(23,26);(5,7,9,12,8,6,10,11)(13,14,19,25,17,21,16,18)(15,20,26,28,23,27,22,24);(2,4)(6,7)(11,12)(13,15,16,22,17,23,19,26)(14,20,18,24,21,27,25,28);(15,23)(20,27)(22,26)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^c=dh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72215867, #72215902, #72215911, … |
| maximal normal subgroup of | #72215867, #72215902, #72215911, … |
| maximal quotient of | #72322745 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109757);