Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(7,11)(10,12)(13,14,18,22,19,23,17,25)(15,24,26,28,27,16,21,20);(6,9)(7,11)(13,15)(14,20)(16,22)(17,26)(18,21)(19,27)(23,28)(24,25);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,16,19,24)(14,21,23,26)(15,25,27,22)(17,28,18,20);(6,9)(10,12)(13,14,18,22,19,23,17,25)(15,16,26,20,27,24,21,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,23,25)(15,21,27,26)(16,20,24,28)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249598, #72249631, #72281349, … |
| maximal normal subgroup of | #72249598, #72249631, #72281349, … |
| maximal quotient of | #72264659, #72264679, #72264712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7102853);