Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,19)(15,21,24,20)(16,25,26,17)(22,23,28,27);(6,9)(10,11)(14,19)(16,26)(20,21)(22,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,22)(17,23)(18,24)(19,21)(25,27)(26,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,17)(15,22)(18,26)(19,25)(20,23)(21,27)(24,28);(13,15)(14,21)(16,22)(17,27)(18,24)(19,20)(23,25)(26,28);(13,17)(14,16)(15,23)(18,25)(19,26)(20,22)(21,28)(24,27)Polycyclic presentation:
2^9|a^2=i,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh140 conjugacy classes — view the full character table.
| maximal subgroup of | #72236842, #72238149, #72238254, … |
| maximal normal subgroup of | #72236842, #72238149, #72238254, … |
| maximal quotient of | #72263774, #72263798, #72297185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534620);