Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573549 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3169 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)(3,4)(13,14)(15,25)(16,28)(17,23)(18,24)(19,22)(20,26)(21,27);(2,4)(6,10)(8,11)(13,15)(14,20)(16,17)(18,26)(19,27)(21,24)(22,28)(23,25);(5,6)(7,8)(9,10)(11,12)(13,16,19,26)(14,21,22,25)(15,17,27,18)(20,24,28,23);(5,7,9,12)(6,11,10,8)(14,22)(20,28)(21,25)(23,24);(5,8,9,11)(6,7,10,12)(13,17,19,18)(14,23,22,24)(15,16,27,26)(20,25,28,21)Polycyclic presentation:
2^9|c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72305357, #72305361, #72305373 |
| maximal normal subgroup of | #72305357, #72305361, #72305373 |
| maximal quotient of | #72305719, #72305740, #72305757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524508);