Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2017 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)(5,6)(7,10)(8,11)(9,12)(13,14,17,16,18,21,26,27)(15,22,25,23,24,20,28,19);(5,6)(7,10)(8,11)(9,12)(13,15,18,24)(14,19,21,23)(16,20,27,22)(17,28,26,25);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,15)(14,20)(16,19)(17,25)(18,24)(21,22)(23,27)(26,28);(5,8,10,9)(6,11,7,12)(13,16,26,14,18,27,17,21)(15,23,28,22,24,19,25,20);(5,9,10,8)(6,11,7,12)(14,21)(15,24)(16,27)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=fh,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308141, #72308416, #72308856 |
| maximal normal subgroup of | #72308141, #72308416, #72308856 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106343);