Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3219 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)(13,14,16,19,18,21,17,22)(15,23,26,20,25,28,24,27);(2,4)(5,6,8,9)(7,10,11,12)(14,19)(16,17)(20,23)(21,22)(24,26)(27,28);(6,9)(7,11)(13,15)(14,20)(16,24)(17,26)(18,25)(19,23)(21,27)(22,28);(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,19,21,22)(15,24,25,26)(20,23,27,28);(6,9)(7,11)(14,21)(19,22)(20,27)(23,28)Polycyclic presentation:
2^9|a^2=fi,b^2=h,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257444, #72257516, #72257525, … |
| maximal normal subgroup of | #72257444, #72257516, #72257525, … |
| maximal quotient of | #72264309, #72264602 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503722);