Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(9,10)(11,12)(13,14,18,19,17,20,16,15);(5,6)(7,8)(14,19)(15,20)(16,18);(2,4)(6,8)(13,14,16,15,17,20,18,19);(10,12)(13,15,18,14,17,19,16,20)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239676, #72239726, #72239754, … |
| maximal normal subgroup of | #72239676, #72239726, #72239754, … |
| maximal quotient of | #72250385, #72250585, #72320421 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327737);