Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7939 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,18)(16,23)(17,20)(19,24)(21,25)(22,28)(26,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(15,21)(18,25)(22,27)(26,28);(5,6,9,12)(7,8,11,10)(13,14,17,20)(15,18,21,25)(16,19,24,23)(22,26,27,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28);(5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27);(5,8)(6,11)(7,12)(9,10)(13,16)(14,19)(15,22)(17,24)(18,26)(20,23)(21,27)(25,28)Polycyclic presentation:
2^9|a^2=g,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72256737, #72297126, #72326380, … |
| maximal normal subgroup of | #72256737, #72297126, #72326380, … |
| maximal quotient of | #72256764, #72297179, #72297385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7944121);