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 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2424 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,17)(18,20)(22,25)(26,27);(1,3,4,2)(6,9)(7,12)(14,17)(18,19)(21,22)(23,27)(24,25)(26,28);(6,9)(10,11)(15,20)(18,19)(22,25)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,16,20)(14,18,17,19)(21,22)(23,26)(24,25)(27,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^2=fgh,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 | #72281715, #72281811, #72281968, … |
| maximal normal subgroup of | #72281715, #72281811, #72281968, … |
| maximal quotient of | #72330767 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 469021);