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 | |
8762 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)(6,9)(10,11)(13,14)(15,23)(16,19)(17,21)(18,22)(20,24)(25,27)(26,28);(1,2)(3,4)(13,15)(14,20)(16,23)(17,22)(18,25)(19,24)(21,28)(26,27);(5,6)(7,11)(8,9)(10,12)(14,16)(15,24)(21,25)(22,26);(5,7,8,12)(6,10,9,11)(13,16,19,14)(15,23,24,20)(17,25,27,21)(18,26,28,22);(13,17)(14,21)(15,22)(16,25)(18,23)(19,27)(20,28)(24,26);(13,18)(14,22)(15,25)(16,26)(17,20)(19,28)(21,24)(23,27)Polycyclic presentation:
2^9|d^2=h,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 | #72238157, #72238166, #72238195, … |
| maximal normal subgroup of | #72238157, #72238166, #72238195, … |
| maximal quotient of | #72297399, #72297411, #72297614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7602108);