Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2053 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)(7,11)(10,12)(13,14,19,20,18,22,17,16)(15,23,28,27,25,21,26,24);(6,9)(7,11)(14,20)(16,22)(17,19)(21,24)(23,27)(26,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,27)(17,26)(18,25)(19,28)(20,24)(22,23);(6,9)(10,12)(13,16,19,14,18,20,17,22)(15,24,28,23,25,27,26,21);(5,7,8,11)(6,10,9,12)(15,25)(21,23)(24,27)(26,28)Polycyclic presentation:
2^9|a^2=fg,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72257063, #72257118, #72257158, … |
| maximal normal subgroup of | #72257063, #72257118, #72257158, … |
| maximal quotient of | #72264640 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7058303);