Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(10,11)(13,14)(15,19)(16,20)(17,18)(21,22,25,27)(23,24,28,26);(5,6)(7,11)(8,9)(10,12)(13,15,17,20)(14,16,18,19)(21,23)(22,26)(24,27)(25,28);(1,2,3,4)(6,9)(10,11)(14,18)(16,19)(22,27)(23,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=i,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232613, #72232619, #72234588, … |
| maximal normal subgroup of | #72232613, #72232619, #72234588, … |
| maximal quotient of | #72237749, #72328380, #72329847 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 483356);