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 | |
8184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6,9,7)(8,11,12,10)(13,14,19,21)(15,18,24,20)(16,27,26,22)(17,28,25,23);(1,2,3,4)(13,15)(14,20)(16,25)(17,26)(18,21)(19,24)(22,23)(27,28);(6,7)(10,11)(14,21)(15,24)(16,26)(23,28);(5,7,9,6)(8,11,12,10)(13,16,19,26)(14,22,21,27)(15,25,24,17)(18,28,20,23);(5,8,9,12)(6,10,7,11)(13,17,19,25)(14,23,21,28)(15,26,24,16)(18,27,20,22);(13,18)(14,24)(15,21)(16,28)(17,22)(19,20)(23,26)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72238147, #72238156, #72238222, … |
| maximal normal subgroup of | #72238147, #72238156, #72238222, … |
| maximal quotient of | #72326300, #72326397, #72326419, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7850429);