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 | |
8399 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)(5,6,9,11)(7,8,12,10)(13,14,19,21)(15,17,16,20)(18,22,27,28)(23,25,24,26);(1,2)(3,4)(13,15,19,16)(14,20,21,17)(18,23,27,24)(22,26,28,25);(5,7,9,12)(6,10,11,8)(14,21)(15,16)(22,28)(23,24);(5,7)(6,8)(9,12)(10,11)(13,16,19,15)(14,17,21,20)(18,24,27,23)(22,25,28,26);(5,8)(6,7)(9,10)(11,12)(13,17)(14,15)(16,21)(18,26)(19,20)(22,24)(23,28)(25,27);(13,18)(14,22)(15,23)(16,24)(17,25)(19,27)(20,26)(21,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=i,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 | #72238165, #72238198, #72238234, … |
| maximal normal subgroup of | #72238165, #72238198, #72238234, … |
| maximal quotient of | C2×#7940421 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7847617);