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 | |
8102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,22)(17,21)(18,20)(23,26)(24,28)(25,27);(1,2)(3,4)(13,15,18,22)(14,19,20,16)(17,23,25,28)(21,26,27,24);(6,9)(10,11)(14,20)(15,22)(17,25)(24,26);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,22)(17,21)(18,20)(23,26)(24,28)(25,27);(5,7)(6,10)(8,12)(9,11)(13,16,18,19)(14,15,20,22)(17,24,25,26)(21,23,27,28);(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72256600, #72256602, #72256656, … |
| maximal normal subgroup of | #72256600, #72256602, #72256656, … |
| maximal quotient of | #72256708, #72256718, #72263428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7701546);