Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(14,19)(15,20)(16,17);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,14,16,19,18,20,17,15);(6,10)(7,11)(13,15,17,20,18,19,16,14);(7,11)(9,12);(14,20)(15,19)Polycyclic presentation:
2^9|b^2=f,c^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72258121, #72282156, #72282571, … |
| maximal normal subgroup of | #72258121, #72282156, #72282571, … |
| maximal quotient of | #72306200, #72306205 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366954);