Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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 | #59412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2594 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)(13,14,19,27,18,21,17,16)(15,22,28,20,23,26,25,24);(6,9)(7,11)(13,15)(14,20)(16,26)(17,28)(18,23)(19,25)(21,24)(22,27);(1,2,3,4)(5,6,8,9)(7,10,11,12)(14,21)(15,23)(16,27)(25,28);(6,9)(10,12)(13,16,17,21,18,27,19,14)(15,24,25,26,23,20,28,22);(5,7,8,11)(6,10,9,12)(14,21)(15,23)(16,27)(25,28)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=f,e^2=hi,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^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249615, #72249628, #72264339, … |
| maximal normal subgroup of | #72249615, #72249628, #72264339, … |
| maximal quotient of | #72265062, #72265154, #72265600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7102903);