Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,13,14,19)(8,11)(9,12)(10,15,18,16)(17,20);(10,18)(13,19);(5,7)(6,10)(8,16)(9,14)(11,19)(12,18)(13,20)(15,17);(2,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^9|a^2=eh,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216067, #72219641, #72219643, … |
| maximal normal subgroup of | #72216067, #72219641, #72219643, … |
| maximal quotient of | #72233071, #72262391 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356777);