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 | |
9053 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)(5,6,8,10)(7,9,12,11)(13,14)(15,23)(16,21)(17,22)(18,20)(19,24)(25,27)(26,28);(1,2)(3,4)(13,15)(14,19)(16,25)(17,21)(18,24)(20,23)(22,26)(27,28);(5,6)(7,11)(8,10)(9,12)(14,20)(15,24)(21,28)(22,25);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,25)(17,21)(18,24)(20,23)(22,26)(27,28);(13,16)(14,21)(15,25)(17,19)(18,26)(20,28)(22,24)(23,27);(13,17)(14,22)(15,21)(16,23)(18,27)(19,26)(20,25)(24,28)Polycyclic presentation:
2^9|a^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72256673, #72256674, #72256676, … |
| maximal normal subgroup of | #72256673, #72256674, #72256676, … |
| maximal quotient of | #72297368, #72297441, #72297615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7601403);