Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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 | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(10,15)(13,16)(17,18)(19,20)(21,23)(22,24);(6,8)(9,10)(11,16)(12,13)(14,15)(17,19,21,24)(18,20,23,22);(5,6)(7,8)(9,11,14,12)(10,16,15,13)(17,19)(18,22)(20,23)(21,24);(1,3,4,2)(9,12,14,11)(10,16,15,13);(9,13,14,16)(10,12,15,11)(17,20,21,22)(18,19,23,24)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=fi,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef140 conjugacy classes — view the full character table.
| maximal subgroup of | #72279775, #72279836, #72280386, … |
| maximal normal subgroup of | #72279775, #72279836, #72280386, … |
| maximal quotient of | C2×#519350 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 426546);