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 | |
8150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14,19,21)(15,20,16,22)(17,28,23,18)(24,27,26,25);(1,2,3,4)(13,15)(14,20)(16,19)(17,24)(18,25)(21,22)(23,26)(27,28);(6,7)(10,11)(14,21)(15,16)(17,23)(25,27);(5,7,9,6)(8,11,12,10)(13,16,19,15)(14,22,21,20)(17,26,23,24)(18,27,28,25);(5,8,9,12)(6,10,7,11)(13,17,19,23)(14,18,21,28)(15,24,16,26)(20,25,22,27);(13,18)(14,23)(15,25)(16,27)(17,21)(19,28)(20,26)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72263085 |
| maximal normal subgroup of | #72263085 |
| maximal quotient of | #72263098 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7620398);