Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,17,21,18,22,25,27)(15,23,16,20,24,28,26,19);(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,22)(21,28)(23,27);(2,4)(5,7,8,12)(6,9,10,11)(15,24)(16,26)(19,20)(23,28);(7,12)(9,11)(13,16,25,15,18,26,17,24)(14,20,27,23,22,19,21,28);(6,10)(9,11)(15,24)(16,26)(19,20)(23,28)Polycyclic presentation:
2^9|a^2=fg,c^2=hi,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972049);