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 | |
8642 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)(13,14,18,17)(15,19,21,23)(16,24,26,20)(22,28,25,27);(1,2)(3,4)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27);(6,9)(10,11)(14,17)(15,21)(20,24)(22,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,20)(15,22)(17,24)(18,26)(19,27)(21,25)(23,28);(13,17,18,14)(15,23,21,19)(16,24,26,20)(22,28,25,27)Polycyclic presentation:
2^9|a^2=gi,f^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 | #72238149, #72238160, #72238245, … |
| maximal normal subgroup of | #72238149, #72238160, #72238245, … |
| maximal quotient of | #72281221, #72281233, #72297143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7706762);