Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994171 |
| 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×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(7,11)(10,12)(13,14,17,22)(15,23,24,19)(16,21,18,20)(25,27,26,28);(2,3)(6,9)(7,11)(13,15)(14,19)(16,26)(17,24)(18,25)(20,28)(21,27)(22,23);(5,6)(7,10)(8,9)(11,12)(14,20)(15,24)(16,18)(19,27)(21,22)(23,28);(1,3,4,2)(6,9)(10,12)(13,16,17,18)(14,21,22,20)(15,25,24,26)(19,28,23,27);(5,7,8,11)(6,10,9,12)(14,22)(15,24)(20,21)(25,26)Polycyclic presentation:
2^9|a^2=f,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72282919, #72283507, #72283930 |
| maximal normal subgroup of | #72282919, #72283507, #72283930 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7052771);