Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | #1999 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,10,9,11,12,7)(14,18,21,25)(15,22)(20,27,28,23)(24,26);(6,7)(8,12)(10,11)(13,14,19,18,17,21,16,25)(15,23,26,28,22,27,24,20);(2,4)(15,22)(20,28)(23,27)(24,26);(5,7,8,6,9,10,12,11)(13,15,16,24,17,22,19,26)(14,20,18,23,21,28,25,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh,d^c=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318326, #72318330, #72318357, … |
| maximal normal subgroup of | #72318326, #72318330, #72318357, … |
| maximal quotient of | #72318586, #72318831, #72318988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 137134);