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 | |
3403 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)(13,14,19,21,18,15,17,22)(16,20,26,27,24,23,25,28);(6,10)(8,11)(13,15,18,14)(16,23,24,20)(17,22,19,21)(25,28,26,27);(2,4)(5,6)(7,8)(9,10)(11,12)(13,16,18,24)(14,20,15,23)(17,25,19,26)(21,27,22,28);(5,7,9,12)(6,11,10,8)(16,24)(20,23)(25,26)(27,28);(5,8,9,11)(6,7,10,12)(13,17,18,19)(14,21,15,22)(16,25,24,26)(20,27,23,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72249712, #72251174, #72251224, … |
| maximal normal subgroup of | #72249712, #72251174, #72251224, … |
| maximal quotient of | #72249714, #72249720, #72264309, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524845);