Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,15)(16,17,20,18);(1,2,3,4)(7,8)(10,12)(13,15,17,20,19,14,18,16);(5,7,9,8)(6,10,11,12)(13,16,18,14,19,20,17,15);(5,8,9,7)(6,10,11,12);(13,17,19,18)(14,20,15,16)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72224734, #72282824, #72282937, … |
| maximal normal subgroup of | #72224734, #72282824, #72282937, … |
| maximal quotient of | #72224719, #72224723, #72305750 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370393);